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$i u_t + \Delta u + V(\epsilon x ) |u|^{p-1} u = 0, \quad x \in \mathbf R^n$

with the critical power $ p = 1 + 4/n, n \ge 2, $ under certain conditions on the inhomogeneous term $ V $ with a small $ \epsilon > 0. $ We also demonstrate that these localized standing-waves converge to standing waves of the nonlinear Schrödinger equation with the homogeneous nonlinearity.

Suppliers always provide free-shipping for retailers whose total order value exceeds or equals an explicit promotion threshold. This paper incorporates a shipping fee in the discrete multi-period newsvendor problem and applies Weak Aggregating Algorithm (WAA) to offer explicit online ordering strategy. It further considers an extended case with salvage value and shortage cost. In particular, online ordering strategies are derived based on return loss function. Numerical examples serve to illustrate the competitive performance of the proposed ordering strategies. Results show that newsvendors' cumulative return losses are affected by the threshold of the order value-based free-shipping. Moreover, the introduction of salvage value and shortage cost greatly improves the competitive performance of online ordering strategies.

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