Application of the GLM technique to a production planning problem

Generalized Lagrange Multipliers (GLM) are used to develop an algorithm for a type of multiproduct single period production planning problem which involves discontinuities of the fixed charge variety. Several properties of the GLM technique are developed for this class of problems and from these properties an algorithm is obtained. The problem of resolving the gaps which are exposed by the GLM procedure is considered, and an example involving a quadratic cost function is explored in detail.     

Sequencing many jobs on a multi-purpose facility

Suppose a given set of jobs has to be processed on a multi-purpose facility which has various settings or states. There is a choice of states in which to process a job and the cost of processing depends on the state. In addition, there is also a sequence-dependent changeover cost between states. The problem is then to schedule the jobs, and pick an optimum setting for each job, so as to minimize the overall operating costs. A dynamic programming model is developed for obtaining an optimal solution to the problem. The model is then extended using the method of successive approximations with a view to handling large-dimensioned problems. This extension yields good (but not necessarily optimal) solutions at a significant computational saving over the direct dynamic programming approach.     

On queues with state-dependent Erlang service

Some general results are derived for single-channel queues with Poisson input and state-dependent Erlang service times in view of the possible use of this model to approximate arbitrary M/G/1-like state-dependent queues in a manner similar to that suggested by Rosenshine, and by Kendall, and Kotiah, Thompson, and Waugh for the M/G/1. Numerical procedures are indicated for the evaluation of stationary state probabilities, expected system sizes and waiting times, and parameter estimation.     

Operations analysis during the underwater search for Scorpions

This paper discusses the operations analysis in the underwater search for the remains of the submarine Scorpion The a priori target location probability distribution for the search was obtained by monte-carlo procedures based upon nine different scenarios concerning the Scorpion loss and associated credibility weights. These scenarios and weights were postulated by others. Scorpion was found within 260 yards of the search grid cell having the largest a priori probability Frequent computations of local effectiveness probabilities (LEPs) were carried out on scene during the search and were used to determine an updated (a posteriori) target location distribution. This distribution formed the basis for recommendation of the current high probability areas for search The sum of LEPs weighted by the a priori target location probabilities is called search effectiveness probability (SEP) and was used as the overall measure of effectiveness for the operation. SEP and LEPs were used previously in the Mediterranean H-bomb search On-scene and stateside operations analysis are discussed and the progress of the search is indicated by values of SEP for various periods during the operation.     

Optimal control of multi-channel service systems

D. P. Heyman, M. Sobel, and M. J. Magazine among others have shown existence of an optimal policy for control of single server queuing systems. For queues under periodic review existence of an analogous rule is established for multi-server systems. Formulation as a dynamic programming problem is given and proofs for existence are presented for finite horizon, infinite horizon and average cost criteria.     

Statistical concepts in computational mathematics

For a young science and even for most old ones, the intensity and variety of activities involved in computer science are unsurpassed. In an effort to provide the extremely varied training needed in the field, curriculum planners have tended to slight probability and statistics. Rarely has it been a requisite and only occasionally a desired elective However, not only is an adequate foundation in probability and statistics necessary for such external computer applications as tracking and other data reduction, Monte Carlo techniques, gaming, operations research, traffic analyses, etc., but it also plays an important role in internal applications to computer science and technology This effort in Rand's continuing study of computational mathematics and its applications offers a number of examples in various areas of computational mathematics and computing machine arithmetic-to say nothing of computer design, time sharing, and the like-to support the thesis that probability and statistics should be a requisite, not simply an elective or, even worse, ignored, at any institution offering a program in computer science. Further more, a potentially profitable source of research problems can be found in the interface between probability and statistics and computer science.     

On a sequential rule for estimating the location parameter of an exponential distribution

Let us assume that observations are obtained at random and sequentially from a population with density function In this paper we consider a sequential rule for estimating μ when σ is unknown corresponding to the following class of cost functions In this paper we consider a sequential rule for estimating μ when σ is unknown corresponding to the following class of cost functions Where δ(X_I,…,X_N) is a suitable estimator of μ based on the random sample (X₁,…, X_N), N is a stopping variable, and A and p are given constants. To study the performance of the rule it is compared with corresponding “optimum fixed sample procedures” with known σ by comparing expected sample sizes and expected costs. It is shown that the rule is “asymptotically efficient” when absolute loss (p=-1) is used whereas the one based on squared error (p = 2) is not. A table is provided to show that in small samples similar conclusions are also true.     

Allocation of carrier-based attack aircraft using non-linear programming

The paper presents the formulation and several solutions of a model for allocating a fixed number of aircraft to carriers and to missions. The amount of damage that can be inflicted is maximized. A nonseparable concave nonlinear objective function expresses diminishing marginal damage. Linear constraints on aireraft, carrier space, and aircraft availability for missions are included. The model is solved using the sequential unconstrained minimization technique (SUMT). The model is presented in terms of a scenario. Several different exponential damage functions are treated, and S-shaped damage functions are discussed.     

A note on adaptive boiler tube pulling

This paper develops an adaptive algorithm for determining boiler tube pulling strategies by postulating a Beta prior on the probability that an individual tube is defective. This prior is updated according to Bayes' Rule as a result of the sample obtained during the tube pulling process.