Allocation of resources to randomly occurring opportunities

An allocation problem is considered in lvhich different kinds of resources must be allocated to various activities, within a given time period. The opportunities for allo'cation appear randomly during this period. Certain assumptions about the values of possible allocations and the distribution of occurrences of opportunities lead to a dynamic programming formulation of the problem. This leads to a system of ordinary differential equations which are (in theory) solvable recursively, and can be solved numerically to any desired degree of precision. An example is given for the allocation of aircraft-carried weapons to targets of opportunity.     

A Lanchester-type model of combat with stochastic rates

The effects of environmental stochasticity in a Lanchester-type model of combat are investigated. The methodology is based on a study of stochastic differential equations with random parameters characterized by dichotomous Markov processes. Exact expressions for the Laplace transforms of the time evolution of the first- and second-order moments of the system are obtained. A special case when the fluctuations in the parameters occur with great rapidity in comparison with the natural time scale of the system is also analyzed. The stochastic stability in the mean-square sense is discussed by using the Routh–Hurwitz criterion and it is found that the stochastic perturbations tend to destabilize the system.     

A flow-shop problem with sequence-dependent additive setup times

The flow-shop scheduling problem with sequence-dependent additive setup times is considered as a special case of the general problem, and a polynomially bounded approximate method is developed to find a minimum makespan permutation schedule. The approximate algorithm is shown to yield optimal results for the two-machine case. A version of Sule's model is defined that produces the first approximation of the optimal solution for this problem. Computational experience along with numerical examples are provided to test the effectiveness of the method.     

A note on the influence of missing operations on scheduling problems

This paper attempts to resolve the existing confusion concerning missing operations. Scheduling problems are classified in two groups: (i) null-continuous (NC)—comprising the problems where an optimal schedule remains optimal on replacement of arbitrarily small processing times (existing operations) with zeros (missing operations); (ii) null-discontinuous (NDC)—comprising those problems which are not null-continuous.     

Criteria and approximate methods for path-constrained moving-target search problems

A search is conducted for a target moving in discrete time among a finite number of cells according to a known Markov process. The searcher must choose one cell in which to search in each time period. The set of cells available for search depends upon the cell chosen in the last time period. The problem is to find a search path, i.e., a sequence of search cells, that either maximizes the probability of detection or minimizes the mean number of time periods required for detection. The search problem is modelled as a partially observable Markov decision process and several approximate solutions procedures are proposed. © 1995 John Wiley & Sons, Inc.     

On the number of games played in a best of (2n – 1) series

Let p(⩾0.5) denote the probability that team A beats B in a single game. The series continues until either A or B wins n games. Assuming that these games are independent replications, we study some features of the distribution of X_n, the number of games played in the series. It is shown that X_n is unimodal, has an IFRA distribution, and is stochastically decreasing in p. Close approximations to its mode, mean, and variance are given. Finally, it is shown that the maximum-likelihood estimator of p based on X_n is unique.     

System availability in a gamma alternating renewal process

We measure the effectiveness of a repairable system by the proportion of time the system is on, where on-time and off-times are assumed independent and both gamma-distributed. This measure is helpful for system planning and control in the short term, before the steady-state is reached, and its mean value is intermediary between instantaneous and steady-state availabilities. We also present other significant results concerning the Gamma Alternating Renewal Process. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 822–844, 1999     

Simulation of stochastic activity networks using path control variates

This article details several procedures for using path control variates to improve the accuracy of simulation-based point and confidence-interval estimators of the mean completion time of a stochastic activity network (SAN). Because each path control variate is the duration of the corresponding directed path in the network from the source to the sink, the vector of selected path controls has both a known mean and a known covariance matrix. This information is incorporated into estimation procedures for both normal and nonnormal responses. To evaluate the performance of these procedures experimentally, we examine the bias, variance, and mean square error of the controlled point estimators as well as the average half-length and coverage probability of the corresponding confidence-interval estimators for a set of SANs in which the following characteristics are systematically varied: (a) the size of the network (number of nodes and arcs); (b) the topology of the network; (c) the percentage of activities with exponentially distributed durations; and (d) the relative dominance of the critical path. The experimental results show that although large improvements in accuracy can be achieved with some of these procedures, the confidence-interval estimators for normal responses may suffer serious loss of coverage probability in some applications.