Estimation of distribution algorithms (EDAs) is a subset of evolutionary algorithms widely used in various optimization problems, known for their favorable results. Each generation of EDAs builds a probabilistic model to represent the most promising individuals, and the next generation is created by sampling from this model. The primary challenge in designing such algorithms lies in effectively constructing the probabilistic model. The mutual exclusivity constraint imposes an additional challenge for EDAs to approach permutation-based problems. In this study, we propose a new EDA called Position-Guided Sampling Estimation of Distribution Algorithm (PGS-EDA) specifically designed for permutation-based problems. Unlike conventional approaches, our algorithm focuses on the positions rather than the elements during the sampling phase. We evaluate the performance of our algorithm on the Permutation Flow-shop Scheduling Problem (PFSP). The experiments conducted on various sizes of Taillard instances provide evidence of the effectiveness of our algorithm in addressing the PFSP, particularly for small and medium-sized problems. The comparison results with other EDAs designed to handle permutation problems demonstrate that our PSG-EDA algorithm consistently achieves the lowest Average Relative Percentage Deviation (ARPD) values in 19 out of the 30 instances of sizes 20 and 50 used in the study. These findings validate the superior performance of our algorithm in terms of minimizing the makespan criterion of the PFSP.

We present an analytical treatment of the path integral formalism for the propagator of the hyperbolic Pöschl-Teller (HPT) potential within a three-parameter generalization of the Dunkl derivative framework. This formulation is achieved using a Pekeris-type approximation for the inverse quadratic term, which arises from the generalized Dunkl derivative. The energy eigenvalues and corresponding wave functions are then precisely determined through both even and odd solutions.

We investigate the one-dimensional nonrelativistic Schrödinger equation within the Jacobi–Dunkl framework. Although the Jacobi–Dunkl operator is well established in harmonic analysis and special-function theory, its systematic use in quantum-mechanical spectral and scattering problems remains comparatively limited. In this work, we develop an explicit parity-resolved formulation in which the reflection-induced deformation is incorporated directly at the operator level. The Jacobi–Dunkl momentum operator, Hamiltonian, weighted scalar product, and parity decomposition are constructed consistently in the associated weighted Hilbert space. For reflection-invariant potentials, the stationary Schrödinger equation is naturally separated into even and odd sectors. The odd sector acquires an additional effective hyperbolic contribution generated by the reflection part of the operator, whereas the even sector is governed by a different reduced equation. The formalism is first applied to the one-dimensional infinite quantum well, for which exact eigenfunctions and parity-dependent quantization conditions are obtained in terms of Jacobi-type hypergeometric functions. We then treat the hyperbolic Pöschl–Teller potential. In the bound-state sector, finite parity-dependent discrete spectra and normalized eigenfunctions expressed through Jacobi polynomials are derived. The corresponding scattering sector is also analyzed by reconstructing the full-line reflection and transmission amplitudes from the even- and odd parity channels. The results show that the Jacobi–Dunkl parameters restructure the spectrum, split the parity branches, modify the bound-state localization, and affect the scattering behavior. The ordinary Schrödinger results are recovered as a formal limiting benchmark, confirming the consistency of the construction.