We analyse the sensitivity of the solution of a nonlinear obstacle plate problem, with respect to small perturbations of the middle plane of the plate. This analysis, which generalizes the results of [9,10] for the linear case, is done by application of an abstract variational result [6], where the sensitivity of parameterized variational inequalities in Banach spaces, without uniqueness of solution, is quantified in terms of a generalized derivative, that is the proto-derivative. We prove that the hypotheses required by this abstract sensitivity result are verified for the nonlinear obstacle plate problem. Namely, the constraint set defined by the obstacle is polyhedric and the mapping involved in the definition of the plate problem, considered as a function of the middle plane of the plate, is semi-differentiable. The verification of these two conditions enable to conclude that the sensitivity is characterized by the proto-derivative of the solution mapping associated with the nonlinear obstacle plate problem, in terms of the solution of a variational inequality.

A new numerical method based on fictitious domain methods for shapeoptimization problems governed by the Poisson equation is proposed.The basic idea is to combine the boundary variation technique, in whichthe mesh is moving during the optimization, and efficient fictitiousdomain preconditioning in the solution of the (adjoint) state equations.Neumann boundary value problems are solved using an algebraic fictitiousdomain method. A mixed formulation based on boundary Lagrangemultipliers is used for Dirichlet boundary problems and the resultingsaddle-point problems are preconditioned with block diagonal fictitiousdomain preconditioners. Under given assumptions on the meshes, thesepreconditioners are shown to be optimal with respect to the conditionnumber. The numerical experiments demonstrate the efficiency ofthe proposed approaches.

This paper derives a good approach to approximating the expected inventory level per unit time for the continuousreview (Q, r) perishable inventory system. Three existing approximation approaches are examined and compared with the proposed approach. Three stockout cases, including the full backorder, the partial backorder, and the full lost sales cases, which customers or material users generally use to respond to a stockout condition are considered.This study reveals the fact that the proposed approximationis simple yet good and suitable for incorporation into the (Q, r)perishable inventory model to determine the best ordering policy.The results from numerical examples and a sensitivity analysisindicate that severe underestimation or overestimation of the expected inventory level per unit time due to the use of an inappropriate approximation approach would result in great distortion in the determination of the best ordering policy.