Staffing decisions for heterogeneous workers with turnover

In this paper we consider a firm that employs heterogeneous workers to meet demand for its product or service. Workers differ in their skills, speed, and/or quality, and they randomly leave, or turn over. Each period the firm must decide how many workers of each type to hire or fire in order to meet randomly changing demand forecasts at minimal expense. When the number of workers of each type can by continuously varied, the operational cost is jointly convex in the number of workers of each type, hiring and firing costs are linear, and a random fraction of workers of each type leave in each period, the optimal policy has a simple hire- up-to/fire-down-to structure. However, under the more realistic assumption that the number of workers of each type is discrete, the optimal policy is much more difficult to characterize, and depends on the particular notion of discrete convexity used for the cost function. We explore several different notions of discrete convexity and their impact on structural results for the optimal policy.     

Single Machine Batch Scheduling Problem with Resource Dependent Setup and Processing Time in the Presence of Fuzzy Due Date

We consider a batch scheduling problem on a single machine which processes jobs with resource dependent setup and processing time in the presence of fuzzy due-dates given as follows:1. There are n independent non-preemptive and simultaneously available jobs processed on a single machine in batches. Each job j has a processing time and a due-date.2. All jobs in a batch are completed together upon the completion of the last job in the batch. The batch processing time is equal to the sum of the processing times of its jobs. A common machine setup time is required before the processing of each batch.3. Both the job processing times and the setup time can be compressed through allocation of a continuously divisible resource. Each job uses the same amount of the resource. Each setup also uses the same amount of the resource.4. The due-date of each job is flexible. That is, a membership function describing non-decreasing satisfaction degree about completion time of each job is defined.5. Under above setting, we find an optimal batch sequence and resource values such that the total weighted resource consumption is minimized subject to meeting the job due-dates, and minimal satisfaction degree about each due-date of each job is maximized. But usually we cannot optimize two objectives at a time. So we seek non-dominated pairs i.e. the batch sequence and resource value, after defining dominance between solutions.A polynomial algorithm is constructed based on linear programming formulations of the corresponding problems.     

Extended structures and new infinite-dimensional hidden symmetries for the Einstein–Maxwell-dilaton-axion theory with multiple vector fields

A so-called extended elliptical-complex (EEC) function method is proposed and used to further study the Einstein–Maxwell-dilaton-axion theory with p vector fields (EMDA-p theory, for brevity) for . An Ernst-like matrix EEC potential is introduced and the motion equations of the stationary axisymmetric EMDA-p theory are written as a so-called Hauser–Ernst-like self-dual relation for the EEC matrix potential. In particular, for the EMDA-2 theory, two Hauser–Ernst-type EEC linear systems are established and based on their solutions some new parametrized symmetry transformations are explicitly constructed. These hidden symmetries are verified to constitute an infinite-dimensional Lie algebra, which is the semidirect product of the Kac–Moody algebra and Virasoro algebra (without centre charges). These results show that the studied EMDA-p theories possess very rich symmetry structures and the EEC function method is necessary and effective.     

On the Path of a Quasi-static Crack in Mode III

A method for finding the path of a quasi-static crack growing in a brittle body is presented. The propagation process is modelled by a sequence of discrete steps optimizing the elastic energy released. An explicit relationship between the optimal growing direction and the parameters defining the local elastic field around the tip is obtained for an anti-plane field. This allows to describe a simple algorithm to compute the crack path. Mathematics Subject Classifications (2000) 74R05, 74B05, 74G70.Gerardo E. Oleaga: Supported by EU-Project Front Singularities University of Leipzig and the Max Planck Institute MIS. Partial support was also provided by the Spanish DGES project BFM2000-0605.     

On the Convergence Behaviour of Variable Stepsize Multistep Methods for Singularly Perturbed Problems

In this note, we investigate the convergence behaviour of linear multistep discretizations for singularly perturbed systems, emphasising the features of variable stepsizes. We derive a convergence result for A()-stable linear multistep methods and specify a refined error estimate for backward differentiation formulas. Important ingredients in our convergence analysis are stability bounds for non-autonomous linear problems that are obtained by perturbation techniques.     

Common complements of two subspaces of a Hilbert space

In this paper we find a necessary and sufficient condition for two closed subspaces, and , of a Hilbert space to have a common complement, i.e. a subspace having trivial intersection with and and such that .Unlike the finite-dimensional case the condition is significantly more subtle than simple equalities of dimensions and codimensions, and non-trivial examples of subspaces without a common complement are possible.     

Safe bounds in linear and mixed-integer linear programming

Current mixed-integer linear programming solvers are based on linear programming routines that use floating-point arithmetic. Occasionally, this leads to wrong solutions, even for problems where all coefficients and all solution components are small integers. An example is given where many state-of-the-art MILP solvers fail. It is then shown how, using directed rounding and interval arithmetic, cheap pre- and postprocessing of the linear programs arising in a branch-and-cut framework can guarantee that no solution is lost, at least for mixed-integer programs in which all variables can be bounded rigorously by bounds of reasonable size. Mathematics Subject Classification (2000):primary 90C11, secondary 65G20     

A superlinearly convergent predictor-corrector method for degenerate LCP in a wide neighborhood of the central path with -iteration complexity

An interior point method for monotone linear complementarity problems acting in a wide neighborhood of the central path is presented. The method has -iteration complexity and is superlinearly convergent even when the problem does not possess a strictly complementary solution. Mathematics Subject Classification (2000): 49M15, 65K05, 90C33 Work supported by the National Science Foundation under Grant No. 0139701. An erratum to this article is available at.     

Early Gravity-Wave Detection Experiments, 1960-1975

I discuss the experiments of Joseph Weber (1919-2000) of the University of Maryland between 1960 and 1973 that were aimed at the detection of gravity waves. He used throughout a quadratic detector to demodulate his antenna signal, which I analyze and compare to a linear detector.The latter was used by all of the other groups that entered the field. Of these, Richard Garwin and I at IBM were one of the first groups to publish, and I discuss in detail our experiments between 1973 and 1975. I then discuss the experiments that were carried out at Bell Labs-Rochester, Glasgow, Munich-Frascati,Moscow, and Tokyo. I compare the results, all of which were negative, with Webers claimed detection of large numbers of gravity-wave events, as many as seven per day. I conclude that these were not in fact gravity waves, but artifacts of his extremely hands-on data-analysis procedures, which I discuss in detail. Finally, I speculate on how this came about.     

On the Existence of Solutions of Quasivariational Inclusion Problems

Quasivariational inclusion problems are formulated and sufficient conditions on the existence of solutions are shown. As special cases, we obtain several results on the existence of solutions of general vector ideal, proper, Pareto, weak quasioptimization problems, quasivariational inequalities, and vector quasiequilibrium problems. Further, we prove theorems on the existence for solutions of systems of these inclusions. As a corollary, we obtain an ideal minimax theorem concerning vector functions.