A problem of restricted partitions

This paper presents a simple algorithm for finding the number of restricted k-partitions of a natural number n. The unrestricted k-partitions of n are expressed as the sum of these restricted k-partitions, called inadmissible, and the admissible k-partitions. The simplicity of the algorithm is striking, though all the implications are unclear.     

Optimal and suboptimal procedures in group sequential sampling

This paper investigates the problem of choosing between two simple hypothesis, H₀ and H₁, in terms of independent, identically distributed random variables, when observations can be taken in groups. At any stage in the decision process it must be decided whether to stop and take action now or to continue, in which case the size of the next group of observations must be decided upon. The problem is to find an optimal procedure incorporating a stopping, group size (batch) and terminal action rule. It is proven, in general, that the optimal stopping and terminal action rule is of the sequential probability ratio type (SPRT). Fixed stopping rules of the SPRT type are studied and an iterative procedure of the policy improvement type, both with and without a value determination step, is developed. It is shown, for the general situation, that both the average risk and scheduling rule converge to the optima. Also, six suboptimal scheduling rules are considered with respect to the average risks they achieve. Numerical results are presented to illustrate the effectiveness of the procedures.     

An informal survey of multi-echelon inventory theory

Contained herein is an informal nonmathematical survey of research in multi-echelon inventory theory covering published results through 1971. An introductory section defines the term, “multi-echelon,” and establishes the kinds of problems involving multi-echelon considerations. Subsequent sections provide surveys of research on deterministic and stochastic multi-echelon inventory control problems, allocation models, and multi-echelon planning and evaluation models. A final section discusses the present state of the art and suggests directions for future research. A bibliography of papers concerning multi-echelon inventory theory and applications is included.     

Impact of an all volunteer force upon the navy in the 1972-1973 timeframe

This paper develops estimates of true volunteer levels for 1972 and 1973, based on experience gained through 1970 draft lottery data. The paper also formulates estimates of the qualitative characteristics of a 1972-1973 Navy volunteer force, and establishes a relationship between rate of volunteerism and military pay. Utilizing estimates generated in the paper, Navy military personnel budget requirements for FY '72 and '73 are presented.     

Quadratic as parametric linear programming

This paper describes an approximate solution procedure for quadratic programming problems using parametric linear programming. Limited computational experience suggests that the approximation can be expected to be “good”.     

On estimation in weibull distributions with random scale parameters

Work by the present authors on life distributions derived from stochastic hazard functions [4] is related to certain articles that have appeared in this journal. This relationship is illustrated. The emphasis of this article is upon problems of parameter estimation.     

Stock control in a many depot system

The chief problems considered are: (1) In a parallel set of warehouses, how should stocks be allocated? (2) In a system consisting of a central warehouse and several subsidiary warehouses, how much stock should be carried in each? The demands may have known, or unknown, distribution functions. For problem (1), the i-th stock n_i should usually be allocated in proportion to the i-th demand m_i; in special cases, a significant improvement is embodied in the formula (N = total allocable stock)

     

A branch and bound algorithm for allocation problems in which constraint coefficients depend upon decision variables

A branch and bound algorithm is developed for a class of allocation problems in which some constraint coefficients depend on the values of certain of the decision variables. Were it not for these dependencies, the problems could be solved by linear programming. The algorithm is developed in terms of a strategic deployment problem in which it is desired to find a least-cost transportation fleet, subject to constraints on men/materiel requirements in the event of certain hypothesized contingencies. Among the transportation vehicles available for selection are aircraft which exhibit the characteristic that the amount of goods deliverable by an aircraft on a particular route in a given time period (called aircraft productivity and measured in kilotons/aircraft/month) depends on the ratio of type 1 to type 2 aircraft used on that particular route. A model is formulated in which these relationships are first approximated by piecewise linear functions. A branch and bound algorithm for solving the resultant nonlinear problem is then presented; the algorithm solves a sequence of linear programming problems. The algorithm is illustrated by a sample problem and comments concerning its practicality are made.     

A maximum utility solution to a vehicle constrained tanker scheduling problem

A modification to the Dantzig and Fulkerson Tanker Scheduling Problem is described. An insufficient number of vehicles and a utility associated with each vehicle delivery are assumed. The new problem is shown to be equivalent to a Transshipment Problem, the solution of which is the same as the maximal utility solution of the modified Tanker Scheduling Problem. An example is given.