This paper formulates six inventory models for products with instantaneous deterioration, focusing on the impacts of full and partial advance payment structures. The demand function depends on both price and stock levels and accounts for shortages through full backlogging. The primary objective is to determine the optimal payment policy under varying trade facilities, analyzing six distinct payment scenarios commonly employed in business practice. Each model is presented with closed-form solutions and supported by mathematical formulations. For each case, algorithms and mathematical proofs are developed to determine the optimal cycle duration and corresponding unit cost. Numerical examples and 2D graphical representations generated using MATLAB are included to validate the proposed models. Additionally, a sensitivity analysis is conducted to examine the effects of each payment policy and parameter variation, providing key managerial insights into payment planning in inventory management.

In wave-guiding structures, a bound state in the continuum (BIC) is a special guided mode with no leakage loss even though lateral radiation channels exist. The BICs are related to high-$Q$ resonances and have found numerous applications in photonics. A particularly simple class of BICs has been found in slab waveguides with an anisotropic core and anisotropic substrate [Gomis-Bresco et al., Nat. Photon. 11, 232 (2017)]. Many of these BICs are robust with respect to structural parameters. In this paper we demonstrate the robustness of BICs for a broad class of anisotropic waveguides of which the core is nonuniform in general. Using a perturbation method, we show that generic BICs persist when the structure is perturbed with any sufficiently small perturbation satisfying some simple conditions. The theory improves the theoretical understanding of BICs in anisotropic media and provides useful guidance for realizing their practical applications.