The asymptotic joint distribution of the state occupancy times in an alternating renewal process

An alternating renewal process starts at time zero and visits states 1,2,…,r, 1,2, …,r 1,2, …,r, … in sucession. The time spent in state i during any cycle has cumulative distribution function F_i, and the sojourn times in each state are mutually independent, positive and nondegenerate random variables. In the fixed time interval [0,T], let U_i(T) denote the total amount of time spent in state i. In this note, a central limit theorem is proved for the random vector (U_i(T), 1 ≤ i ≤ r) (properly normed and centered) as T → ∞.     

Optimal replacement for fault-tolerant systems

We consider the optimal replacement problem for a fault tolerant system comprised of N components. The components are distingushable, and the state of the system is given by knowing exactly which components are operationl and which have failed. The individual component failure rates depend on the state of the entire system. We assume that the rate at which the system produces income decreases as the system deteriorates and the system replacement cost rises. Individual components cannot be replaced. We give a greedy-type algorithm that produces the replacement policy that maximizes the long-run net system income per unit time.     

Multiobjective fractional programming duality theory

Results of Geoffrion for efficient and properly efficient solutions of multiobjective programming problems are extended to multiobjective fractional programming problems. Duality relationships are given for these problems where the functions are generalized convex or invex.     

Parallel machine replacement

We consider the parallel replacement problem in which there are both fixed and variable costs associated with replacing machines. Increasing maintenance costs motivate replacements, and the fixed replacement cost provides incentive for replacing machines of different ages together in “clusters.” We prove two intuitive results for this problem. First, it is never optimal to split a cluster of like-aged machines, and second, it is never optimal to replace newer clusters before older clusters. By incorporating these two results into an algorithmic approach, we vastly reduce the amount of computation required to identify an optimal replacement policy.     

Economic lot scheduling of fully loaded processes with external setups

Without restricting the class of permissible schedules, we derive optimal schedules for economic lot scheduling problems that are fully loaded, have external setups, and have only two products. The fully loaded condition accurately represents certain types of bottlenecks. We show that the optimal schedule must have the Wagner-Whitin property. We also develop a measure of aggregate inventory, derive an optimal steady-state aggregate inventory policy, and provide conditions under which the aggregate inventory level of an optimal schedule must approach a steady state. By restricting the class of permissible schedules to rotation cycle schedules, we extend these results to more than two products.     

A one-dimensional helicopter-submarine game

The article considers a two-person zero-sum game in which a searcher with b bombs wishes to destroy a mobile hider. The players are restricted to move on a straight line with maximum speeds v and u satisfying v > u > 0; neither player can see the other but each knows the other's initial position. The bombs all have destructive radius R and there is a time lag T between the release of a bomb and the bomb exploding. The searcher gets 1 unit if the hider is destroyed and 0 if he survives. A solution is given for b = 1, and extended to b > 1 when the time lag is small. Various applications of the game are discussed.