Evaluation of costs of ordering policies in large machine repair problems

This article examines a version of the machine repair problem where failures may be irreparable. Since the number of machines in the system keeps decreasing, we impose a fixed state-dependent ordering policy of the type often encountered in inventory models. Although the system is Markovian, the number of states becomes very large. The emphasis of the article, therefore, is on deriving computationally tractable formulas for the steady-state probabilities, the long-run average cost per unit time, and the vector of expected discounted costs. When the state space is so large that exact computations may be infeasible, we propose approximations which are relatively quick and simple to compute and which yield very accurate results for the test problems examined.     

Hu's precedence tree scheduling algorithm: A simple proof

This article presents a simple proof of Hu's algorithm for scheduling in minimum time a set of tasks constrained by precedence tree constraints, each task requiring a unit time to complete, and where m processors are available.     

Control charts for exponentially distributed product life

This paper introduces a special control chart procedure for exponentially distributed product life. Statistical control of product life in manufacturing requires continuing life tests of manufactured product so as to detect changes in product life and take appropriate corrective action. These life testing experiments may become exceedingly time consuming and thus can be both impractical because of serious time delays in implementing corrective action on the process when indicated, and quite uneconomical. It is desirable to inquire into the character of life testing by means of a control chart procedure based on the real time to first failure within given samples. Measuring the real minimum life provides a considerable reduction in duration of the testing procedure and in the number of specimens destroyed, yielding a considerable economy over the Shewhart's X control chart.     

A theory of ideal linear weights for heterogeneous combat forces

Detailed combat simulations can produce effectiveness tables which measure the effectiveness of each weapon class on one side of an engagement, battle, or campaign to each weapon class on the other. Effectiveness tables may also be constructed in other ways This paper assumes that effectiveness tables are given and shows how to construct from them a system of weapon weights each of which is a weighted average of the effects of a given weapon against each of the enemy's weapons. These weights utilize the Perron- Frobenius theory of eigenvectors of nonnegative matrices. Methods of calculation are discussed and some interpretations are given for both the irreducible and reducible cases.     

Postoptimality analysis in integer programming by implicit enumeration: The mixed integer case

This paper develops a method for doing postoptimality analysis on the mixed integer programming problem. The proposed procedures form a natural adjunct to enumerative I.P. algorithms that are linear programming based, and they are designed, in effect, to capitalize on insights generated as the problem is initially solved to do subsequent analysis upon it. In particular, limited ranging analysis is possible on selected parameters, as is the efficient resolving of the problem following parameter changes.     

The development and evaluation of a cost-based composite scheduling rule

A cost-based composite scheduling rule is developed and evaluated in comparison with three other well-researched scheduling rules—SPT, S/OPN, and SST. This cost rule permits the optimization of more than one performance measure at a time. The priority number that is used for scheduling operations through each machine group is based on four separate performance measures—(1) In-process Inventory, (2) Facilities Utilization, (3) Lateness, and (4) Mean Setup Time. The factorial experimental design involved three factor levels of loads, three factor levels of cost, and three factor levels of mean time. Analysis of variance was performed on each of the five output measures to study the effects of each of the three factors on each individual rule. Rank-order comparisons between rules were also made; and, finally, general conclusions with regard to the effectiveness and flexibility of the Cost Rule were drawn.     

Some simple scheduling algorithms

This paper considers situations in which jobs require only one operation on a single machine, or on one of a set of identical machines. Penalty-free interruption is allowed. Some simple algorithms are given for finding optimum schedules to minimize maximum lateness and total delay, for the single-machine case, and maximum lateness for a restricted multi-machine case. A simple flow problem formulation permits minimizing maximum lateness for the more general multimachine case.     

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.