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.     

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”.     

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)

     

Suboptimal algorithms for the quadratic assignment problem

The idea of combining relatively simple continuous methods with discrete procedures is used for the construction of suboptimal algorithms for quadratic assignment problems. Depending on the nature of the special problem these steps may vary in complexity. The simplest procedures require minimum storage space and result in tolerable computation times. Different choices of parameters and random variations may be used in order to obtain statistical distributions of suboptimal solutions. Computational results for sample problems indicate improvements on results of Steinberg, Gilmore, and Hillier and Connors.     

A finiteness proof for modified dantzig cuts in integer programming

Let be a basic solution to the linear programming problem subject to: where R is the index set associated with the nonbasic variables. If all of the variables are constrained to be nonnegative integers and x_u is not an integer in the basic solution, the linear constraint is implied. We prove that including these “cuts” in a specified way yields a finite dual simplex algorithm for the pure integer programming problem. The relation of these modified Dantzig cuts to Gomory cuts is discussed.     

Nonlinear programming the choice of direction by gradient projection

Rosen's method of Gradient Projection chooses a search direction which is not necessarily the direction of steepest ascent. However, the projection of the gradient onto a “suitably chosen subspace” does yield the direction of steepest ascent. The suitable choice is easily recognized as a result of some theorems relating gradient projection to steepest ascent. These results lead to a modification of Rosen's method. The modification improves the choice of search direction and usually yields the steepest ascent direction without solving a quadratic programming problem.