A novel approach, referred to as the homotopy stochastic finite element method, is proposed to solve the eigenvalue problem of a structure associated with some amount of uncertainty based on the homotopy analysis method.For this approach, an infinite multivariate series of the involved random variables is proposed to express the random eigenvalue or even a random eigenvector. The coefficients of the multivariate series are determined using the homotopy analysis method. The convergence domain of the derived series is greatly expanded compared with the Taylor series due to the use of an approach function of the parameter h. Therefore, the proposed method is not limited to random parameters with small fluctuation. However, in practice, only singlevariable and double-variable approximations are employed to simplify the calculation. The numerical examples show that with a suitable choice of the auxiliary parameter h, the suggested approximations can produce very accurate results and require reduced or similar computational efforts compared with the existing methods.KEYWORDS homotopy analysis method, perturbation method, random eigenvalue problem, stochastic finite element method, Taylor series 1 | INTRODUCTION Algebraic eigenvalue problems are a class of basic and significant problems in various fields, such as structural dynamics and structural stability. Currently, the computation of eigenvalues and eigenvectors is well comprehended for deterministic problems. 1,2 In many practical cases, however, the physical properties of the structural systems are not deterministic. For instance, the stiffness of a beam can be affected by material imperfections such that the stiffness distribution along the beam is irregular and difficult to measure and the boundary constraints of beam structures are usually uncertain. 3,4 Therefore, it is extremely necessary to use random variables to more realistically describe the uncertain characteristics that exist in eigenvalue problems in engineering.Due to the randomness of the input parameters, such as the modulus of elasticity, of a physical problem, the desired output or eigenvalues will also be random. The methods for computing these random outputs are generally composed of 2 categories. The first category includes simulation-based methods. All orders of statistical moments and all probability density functions of the eigenvalues can be determined by repeatedly carrying out computations of the deterministic problem in the simulation-based methods. Direct Monte-Carlo (DMC) simulation is the most important and fundamental simulation-based method, 5-9 but it requires considerable computational effort, especially for large systems. Even so, DMC simulation can be conducted and is considered a closed-form solution for evaluating approximation methods. The second category for random analysis, stochastic finite element methods (SFEM), 10-13

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We consider pricing problems when customers choose according to the generalized extreme value (GEV) models and the products have the same price sensitivity parameter. First, we consider the static pricing problem, where we maximize the expected profit obtained from each customer. We show that the optimal prices of the different products have a constant markup over their unit costs. We provide an explicit formula for the optimal markup in terms of the Lambert-W function. This result holds for any arbitrary GEV model. Second, we consider single-resource dynamic pricing problems. We show that as we have more resource inventory or as we have fewer time periods left until the end of the selling horizon, the prices charged by the optimal policy decrease. Third, we consider dynamic pricing problems over a network of resources. We focus on a price-based deterministic approximation with prices as the decision variables, but this deterministic approximation fails to be a convex program. We transform the price-based deterministic approximation to an equivalent market-share-based deterministic approximation, with purchase probabilities as the decision variables. Surprisingly, the transformed problem is a convex program, and the gradient of its objective function can be computed efficiently. Computational experiments show that the market-share-based formulation provides substantial advantages over the original price-based formulation.

When retailers decide which assortment of products to offer, they can make use of a choice model that describes how customers choose and substitute among the products. The key is to use a choice model that faithfully captures the choice process of customers, while making sure that the corresponding problem of finding the revenue-maximizing assortment remains tractable. In “Assortment Optimization Under the Paired Combinatorial Logit Model,” Zhang, Rusmevichientong, and Topaloglu consider the paired combinatorial logit model to capture the choice process of customers. This choice model uses a utility maximization framework to capture the customer choices, and the utilities of the products can have a rather general correlation structure. The authors demonstrate that one can construct algorithms with performance guarantees to solve the assortment optimization problem under this choice model.

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