A primal-dual cutting-plane algorithm for all-integer programming

The integer programming literature contains many algorithms for solving all-integer programming problems but, in general, existing algorithms are less than satisfactory even in solving problems of modest size. In this paper we present a new technique for solving the all-integer, integer programming problem. This algorithm is a hybrid (i.e., primal-dual) cutting-plane method which alternates between a primal-feasible stage related to Young's simplified primal algorithm, and a dual-infeasible stage related to Gomory's dual all-integer algorithm. We present the results of computational testing.     

A heterogeneous arrival and service queueing loss model

Consider a single-server exponential queueing loss system in which the arrival and service rates alternate between the paris (γ₁, γ₁), and (γ₂, μ₂), spending an exponential amount of time with rate cμ_i in (γ_i, μ_i), i = 1.2. It is shown that if all arrivals finding the server busy are lost, then the percentage of arrivals lost is a decreasing function of c. This is in line with a general conjecture of Ross to the effect that the “more nonstationary” a Poisson arrival process is, the greater the average customer delay (in infinite capacity models) or the greater the precentage of lost customers (in finite capacity models). We also study the limiting cases when c approaches 0 or infinity.     

Determining confidence bounds for highly reliable coherent systems based on a paucity of component failures

A computationally simple method for obtaining confidence bounds for highly reliable coherent systems, based on component tests which experience few or no failures, is given. Binomial and Type I censored exponential failure data are considered. Here unknown component unreliabilities are ordered by weighting factors, which are firstly presumed known then sensitivity of the confidence bounds to these assumed weights is examined and shown to be low.     

AN ALGORITHM FOR THE SEGREGATED STORAGE PROBLEM

The segregated storage problem involves the optimal distribution of products among compartments with the restriction that only one product may be stored in each compartment. The storage capacity of each compartment, the storage demand for each product, and the linear cost of storing one unit of a product in a given compartment are specified. The problem is reformulated as a large set-packing problem, and a column generation scheme is devised to solve the associated linear programming problem. In case of fractional solutions, a branch and bound procedure is utilized. Computational results are presented.     

Optimal replacement under additive damage and other failure models

A machine or production system is subject to random failure. Upon failure the system is replaced by a new one, and the process repeats. A cost is associated with each replacement, and an additional cost is incurred at each failure in service. Thus, there is an incentive for a controller to attempt to replace before failure occurs. The problem is to find an optimal control strategy that balances the cost of replacement with the cost of failure and results in a minimum total long-run average cost per unit time. We attack this problem under the cumulative damage model for system failure. In this failure model, shocks occur to the system in accordance with a Poisson process. Each shock causes a random amount of damage or wear and these damages accumulate additively. At any given shock, the system fails with a known probability that depends on the total damage accumulated to date. We assume that the cumulative damage is observable by the controller and that his decisions may be based on its current value. Supposing that the shock failure probability is an increasing function of the cumulative damage, we show that an optimal policy is to replace either upon failure or when this damage first exceeds a critical control level, and we give an equation which implicitly defines the optimal control level in terms of the cost and other system parameters. Also treated are some more general models that allow for income lost during repair time and other extensions.     

A model in which component failure rates depend on the working set

We consider a multicomponent system in which the failure rate of a given component at any time depends on the set of working components at that time. Sufficient conditions are presented under which such a system has a life distribution of specified type. The Laplace transform of the time until all components are down is derived. When repair is allowed, conditions under which the resulting process is time reversible are presented.     

Confidence intervals in discrete event simulation: A comparison of replication and batch means

Suppose that we have enough computer time to make n observations of a stochastic process by means of simulation and would like to construct a confidence interval for the steady-state mean. We can make k independent runs of m observations each (n=k.m) or, alternatively, one run of n observations which we then divide into k batches of length m. These methods are known as replication and batch means, respectively. In this paper, using the probability of coverage and the half length of a confidence interval as criteria for comparison, we empirically show that batch means is superior to replication, but that neither method works well if n is too small. We also show that if m is chosen too small for replication, then the coverage may decrease dramatically as the total sample size n is increased.     

The bottleneck transportation problem

The bottleneck transportation problem can be stated as follows: A set of supplies and a set of demands are specified such that the total supply is equal to the total demand. There is a transportation time associated between each supply point and each demand point. It is required to find a feasible distribution (of the supplies) which minimizes the maximum transportaton time associated between a supply point and a demand point such that the distribution between the two points is positive. In addition, one may wish to find from among all optimal solutions to the bottleneck transportation problem, a solution which minimizes the total distribution that requires the maximum time Two algorithms are given for solving the above problems. One of them is a primal approach in the sense that improving fcasible solutions are obtained at each iteration. The other is a “threshold” algorithm which is found to be far superior computationally.     

A queueing system with multiple service time distributions

This paper explores a modification of the output discipline for the Poisson input, exponential output, single channel, first-come, first-served queueing system. Instead, the service time distribution of customers beginning service when alone in the system is considered different from that governing service times of all other customers. More specifically, the service times of lone customers are governed by a one parameter gamma distribution, while the service times of all other customers are exponentially ajstributed. The generating function for the steady-state probsbilities, n_j = Pr { j customers in system at an arbitrary point of departure}, of the imbedded chain, {X_n/X_n = number in system after nth customer is serviced}, is obtained, and the steady-state probabilities, themselves, are found in closed form.