Terms Of Auxiliary Functions Research Articles

Next-day operating room scheduling with uncertain surgery durations: Exact analysis and heuristics

Operating rooms are units of particular interest in hospitals as they constitute more than 40% of total expenses and revenues. Managing operating rooms is challenging due to conflicting priorities and preferences of various stakeholders and the inherent uncertainty of surgery durations. In this study, we consider the next-day scheduling problem of a hospital operating room. Given the list and the sequence of non-identical surgeries to be performed in the next day, one needs to determine the scheduled durations of surgeries where the actual duration of each surgery is uncertain. Our objective is to minimize the weighted sum of expected patient waiting times, room idle time and overtime. First, we provide a reformulation of the objective function in terms of auxiliary functions with a recursive pattern that enables exact analysis of the optimal surgery durations at the expense of high CPU time. Next, we develop and analyze simple-to-use and close-to-optimal scheduling heuristics motivated by practice, for the OR managers to deploy in the field. Our proposed hybrid heuristic attains 1.22% average performance gap and worst average optimality gap of 2.77%. Our solution is easy to implement as it does not require any advanced optimization tool, which is the reality of many operating room environments.

Antiplane crack at the interface of two bonded dissimilar graded piezoelectric materials

The problem of an antiplane crack situated in the interface of two bonded dissimilar graded piezoelectric half-spaces is considered under the permeable crack assumption. The mechanical and electrical properties of the half-spaces are considered for a class of functional forms for which the equilibrium equation has analytic solutions. By using an integral transform technique, the problem is reduced to dual integral equations which are transformed into a Fredholm integral equation by introducing an auxiliary function. The stress intensity factors are obtained in explicit form in terms of auxiliary functions. By solving the Fredholm integral equation numerically, the numerical results for stress intensity factors are obtained which have been displayed graphically to show the influence of the graded piezoelectric materials.

Existence of solutions of boundary-value problems for thin elastic irregular shells

We study the solvability of a geometrically and physically nonlinear boundary-value problem for thin elastic anisotropic irregular shells with clamped edge. For this problem, we suggest a method based on solving the problem in some function space other than the space of displacements and stresses. This method is based on the integral representations of displacement components and strains in terms of auxiliary functions (conventional strains). This method permits obtaining, as the equilibrium equations, nonlinear singular integral equations over a bounded plane domain for the conventional strains. The solvability of these equations is studied by the contraction mapping principle.

Positive solutions for semilinear elliptic equations with singular forcing terms

We consider the existence of solutions to the semilinear elliptic problem (∗) κ { − Δ u + u = u p + κ ∑ i = 1 m c i δ a i in D ′ ( R N ) , u > 0 a.e. in R N and u ( x ) → 0 as | x | → ∞ , with prescribed given finite points { a i } i = 1 m in R N and positive numbers { c i } i = 1 m , where N ⩾ 3 , 1 < p < N / ( N − 2 ) , κ ⩾ 0 is a parameter, and δ a is the Dirac delta function supported at a ∈ R N . We reduce the problem (∗) κ to the problem in H 1 ( R N ) ∩ C 0 ( R N ) in terms of auxiliary functions, and then show the existence of a positive constant κ ∗ > 0 such that (∗) κ has at least two solutions if κ ∈ ( 0 , κ ∗ ) , a unique solution if κ = κ ∗ , and no solution if κ > κ ∗ .

Ab initio studies of structural features not easily amenable to experiment: Part 62. The 4–21G optimized structure of the steroid, 4,4-dimethylandrostan-3-one

The geometry of the chair form of 4,4-dimethylandrostan-3-one (4,4-DMA was determined by ab initio gradient optimization on the 4–21G level. A previously described multiplicative integral approximation (MIA) was used together with the direct SCF approach. In MIA, products of Gaussian functions are expanded in terms of auxiliary functions in order to accelerate the computation of two-electron integrals. Integrals for which the approximation leads to errors exceeding a preset value are systematically corrected, so that MIA assisted calculations yield the same results as conventional methods. Bond distance and angle patterns in the resulting structure of 4,4-DMA are similar to corresponding trends known for smaller systems. In agreement with previous X-ray crystallographic studies, characteristic conformation—transmission effects are found, and the skeleton of the system is bent, i.e. axial bonds at different ends of the molecule are not parallel and there is a non-zero angle between molecular planes in different parts of the system, which would be parallel in an undistorted all- trans skeleton. The calculations determine the steroid structure with a resolution which is currently not afforded by any experimental method operating with unperturbed molecular states, and conformational features can be described without the uncertain effects of packing forces which frequently affect molecular structures in crystals.

Ab initio calculations on large molecules: The multiplicative integral approximation

AbstractIn the multiplicative integral approximation (MIA), two‐electron integrals are evaluated using an expansion of a product of two Gaussians in terms of auxiliary functions. An estimator of the error introduced by the approximation is incorporated in the self‐consistent field (SCF) calculations and the integrals for which the error estimate is larger than a preset value are systematically corrected. In this way the results of a MIA‐assisted calculation have the same accuracy as a conventional calculation. The full exploitation of the expansion technique while constructing the Fock‐matrix allows important time savings. Results are presented for a number of test cases.

Two-port terminations for maximum power gain

The problem of the maximum power gain which can be obtained from an active twoport device has been considered by several authors, notably by Linvill and Schimpf (1956). They calculated the optimum source and load immittances and the resulting maximum power gain, for a given network whose parameters were assumed to satisfy the usual stability condition. Linvill and Schimpf used a somewhat complicated analysis, and expressed the results in terms of auxiliary functions of the two-port parameters. It may therefore be worth pointing out that the problem may be solved simply and explicitly from the wellknown condition, mentioned by Linvill and Schimpf, that the optimum terminations provide a conjugate match at the input and output terminals of the network. In the following analysis, h parameters are used; the expressions can be readily adapted to z, y, or g parameters by means of the usual transformations.