A generalised incomplete beta function and its application to multi-line stock control

The distribution function for the negative binomial distribution is known to be an incomplete beta function. Here, some of the properties of the family of distribution functions for multivariate negative binomial distributions are explored. These properties are then used in deriving the expected cost per unit time for a multi-line joint-reordering system with Poisson demands. Policies are considered for which the quantity of any particular line in stock is the same at the beginning of every cycle. A method which gives good approximations to the optimal values of these quantities is described.

Some problems of stochastic resource allocation

Suppose that a number of jobs are to be carried out by a single processing unit orserver. The server can process any number of jobs at the same time but the time taken to finish all the jobs is the same no matter what scheduling policy is used, provided the server is never left idle. It is, however, possible to minimise the total delays to individual jobs by adopting a suitable policy. Optimal policies are derived here for the situation when the processing times are random variables. If the completion rates (analogous to hazard rates) for every job are increasing it is optimal to process the jobs in sequence. If they are decreasing the optimal policy involves simultaneous processing. These results are applied to a more general resource allocation problem arising in industrial chemical research.

Resource allocation in speculative chemical research

A stochastic theory is given for speculative industrial chemical research. It is shown how the theory may be used to suggest a suitable balance between long-term and short-term projects.

On Bayesian models in stochastic scheduling

The D.A.I. theorem of Gittins and Jones has proved a powerful tool in solving sequential statistical problems. A generalisation of this theorem is presented. This generalisation enables us to solve certain stochastic scheduling problems where the items or jobs to be scheduled have random times to completion, the random times having distributions dependent upon parameters to which prior distributions are allocated. Such problems are of interest in many areas where scheduling is important.

Multiserver scheduling of jobs with increasing completion rates

Under certain conditions it is shown that for jobs with increasing completion rates the expected flow time is minimised by a non-pre-emptive service discipline.

Optimal resource allocation in chemical research

The problem of allocating scientific effort between a number of competing projects is considered. A stochastic mathematical model is set up, and for the case of just two competing projects a number of results are proved which enable the best policy to be determined. An indication is given of the ways in which these results might be extended so as to deal with more general situations.

Optimal selection policies for a sequence of candidate drugs

Pharmaceutical companies have to face huge risks and enormous costs of production before they can produce a drug. Efficient allocation of resources is essential to help in maximizing profits. Yu and Gittins (2007) described a model and associated software for determining efficient allocations for a preclinical research project. This is the starting point for this paper. We provide explicit optimal policies for the selection of successive candidate drugs for two restricted versions of the Yu and Gittins (2007) model. To some extent these policies are likely to be applicable to the unrestricted model.

A forwards induction approach to candidate drug selection

A forwards induction policy is a type of greedy algorithm for Markov decision processes. It is straightforward to implement and is optimal for a large class of models, especially in stochastic resource allocation. In this paper we consider a model for the optimal allocation of resources in pre-clinical pharmaceutical research. We show that although they are not always strictly optimal, forwards induction policies perform well.

A hamiltonian approach to optimal stochastic resource allocation

The problem of scheduling items for service with random service times is formulated as an optimal control problem. Pontryagin's maximum principle is used to determine the optimal schedule in certain cases.

A decision theoretic approach to sample size determination in clinical trials

In this paper, we discuss a Behavioral Bayes approach to the determination of sample size in phase III clinical trials for which the data are assumed to come from a normal distribution for which the mean and variance are both unknown. Software is described which minimizes the expected net cost as a function of the sample size, thereby establishing the optimal sample size. This methodology extends previous work by the assumption of unknown variance. Numerical examples show that the more general model can have a large effect on the optimal sample size, compared with a procedure which uses the known variance model with an estimate of the variance.

Prediction of biological activity for high-throughput screening using binary kernel discrimination

High-throughput screening has made a significant impact on drug discovery, but there is an acknowledged need for quantitative methods to analyze screening results and predict the activity of further compounds. In this paper we introduce one such method, binary kernel discrimination, and investigate its performance on two datasets; the first is a set of 1650 monoamine oxidase inhibitors, and the second a set of 101 437 compounds from an in-house enzyme assay. We compare the performance of binary kernel discrimination with a simple procedure which we call "merged similarity search", and also with a feedforward neural network. Binary kernel discrimination is shown to perform robustly with varying quantities of training data and also in the presence of noisy data. We conclude by highlighting the importance of the judicious use of general pattern recognition techniques for compound selection.

Observations on a result in sequential analysis by P. Whittle

In a series of papers Whittle has proposed locating the optimal boundary between the continuation region and the stopping region, in the no-overshoot approximation, for a class of sequential analysis problems by using the criterion that the gradient of the objective function must be continuous at the boundary. It is shown here that the result originally proposed by Whittle is not true in the generality which he supposed, and an indication is given of some of the circumstances in which it does hold.