An exact ceiling point algorithm for general integer linear programming

We present an algorithm called the exact ceiling point algorithm (XCPA) for solving the pure, general integer linear programming problem (P). A recent report by the authors demonstrates that, if the set of feasible integer solutions for (P) is nonempty and bounded, all optimal solutions for (P) are “feasible 1-ceiling points,” roughly, feasible integer solutions lying on or near the boundary of the feasible region for the LP-relaxation associated with (P). Consequently, the XCPA solves (P) by implicitly enumerating only feasible 1-ceiling points, making use of conditional bounds and “double backtracking.” We discuss the results of computational testing on a set of 48 problems taken from the literature.     

Bounds for circular error probabilities

Bounds for P(X + X ⩽ k²σ) are given where X₁ and X₂ are independent normal variables having zero means and variances σ, σ, respectively. This is generalized when X₁ and X₂ are dependent variables with known covariance matrix.     

Optimal sequential replenishment of ships during combat

A carrier battle group is operating in an area where it is subject to attack by enemy aircraft. It is anticipated that air raids will occur in large waves. The uncertain time between raids is available for the replenishment of supplies. We consider the problem of how best to schedule ammunition replenishment during this period. The theory of Gittins indices provides the technical background to the development of a range of models which yield a hierarchy of index-based heuristics for replenishment. One such heuristic is assessed computationally in a more realistic scenario than is explicitly allowed for by the models.     

A replacement model with overhauls and repairs

This article generalizes the classical periodical replacement model with minimal repair by considering the effect of the overhaul, which is assumed to be able to rejuvenate the system. The generalization is based on the notion of the virtual age of a system introduced by Kijima, Morimura, and Suzuki [“Periodical Replacement Problem without Assuming Minimal Repair,” European Journal of Operational Research, 37 , 194–203 (1988)]. © 1995 John Wiley & Sons, Inc.     

Makespan minimization for m parallel identical processors

We introduce an algorithm, called TMO (Two-Machine Optimal Scheduling) which minimizes the makespan for two identical processors. TMO employs lexicographic search in conjunction with the longest-processing time sequence to derive an optimal schedule. For the m identical parallel processors problem, we propose an improvement algorithm, which improves the seed solution obtained by any existing heuristic. The improvement algorithm, called Extended TMO, breaks the original m-machine problem into a set of two-machine problems and solves them repeatedly by the TMO. A simulation study is performed to evaluate the effectiveness of the proposed algorithms by comparing it against three existing heuristics: LPT (Graham, [11]), MULTIFIT (Coffman, Garey, and Johnson, [6]), and RMG (Lee and Massey, [17]). The simulation results show that: for the two processors case, the TMO performs significantly better than LPT, MULTIFIT, and RMG, and it generally takes considerably less CPU time than MULTIFIT and RMG. For the general parallel processors case, the Extended TMO algorithm is shown to be capable of greatly improving any seed solution. © 1995 John Wiley & Sons, Inc.     

Optimal force mix in heterogeneous combat

This article presents models for determining the optimum number of Red weapons required to win a heterogeneous combat in which m(m> 1) types of Red weapons face a single type of Blue weapon under a newly defined termination policy. Red aims at either minimizing the total cost or maximizing the aggregated remaining force strength. Kuhn-Tucker and simulated annealing techniques are used for obtaining the optimal solution. The methodology is illustrated by analysing heterogeneous combat to determine (i) the feasibility of introducing new types of weapons and (ii) the number of weapons required to win if a specific type of weapon, say infantry, dominates. © 1995 John Wiley & Sons, Inc.     

Maintenance of a device with age-dependent exponential failures

We consider a device that deteriorates over time according to a Markov process so that the failure rate at each state is constant. The reliability of the device is characterized by a Markov renewal equation, and an IFRA (increasing failure rate on average) property of the lifetime is obtained. The optimal replacement and repair problems are analyzed under various cost structures. Furthermore, intuitive and counterintuitive characterizations of the optimal policies and results on some interesting special problems are presented. © 1992 John Wiley & Sons, Inc.     

Bayesian inference for a Lanchester type combat model

We undertake inference for a stochastic form of the Lanchester combat model. In particular, given battle data, we assess the type of battle that occurred and whether or not it makes any difference to the number of casualties if an army is attacking or defending. Our approach is Bayesian and we use modern computational techniques to fit the model. We illustrate our method using data from the Ardennes campaign. We compare our results with previous analyses of these data by Bracken and Fricker. Our conclusions are somewhat different to those of Bracken. Where he suggests that a linear law is appropriate, we show that the logarithmic or linear‐logarithmic laws fit better. We note however that the basic Lanchester modeling assumptions do not hold for the Ardennes data. Using Fricker's modified data, we show that although his “super‐logarithmic” law fits best, the linear, linear‐logarithmic, and logarithmic laws cannot be ruled out. We suggest that Bayesian methods can be used to make inference for battles in progress. We point out a number of advantages: Prior information from experts or previous battles can be incorporated; predictions of future casualties are easily made; more complex models can be analysed using stochastic simulation techniques. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 541–558, 2000     

Note on “An Optimal Recursive Method for Various Inventory Replenishment Models with Increasing Demand and Shortages” by Teng et al.

In a recent paper, Teng, Chern, and Yang consider four possible inventory replenishment models and determine the optimal replenishment policies for them. They compare these models to identify the best alternative on the basis of minimum total relevant inventory costs. The total cost functions for Model 1 and Model 4 as derived by them are not exact for the comparison. As a result, their conclusion on the least expensive replenishment policy is incorrect. The present article provides the actual total costs for Model 1 and Model 4 to make a correct comparison of the four models. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 602–606, 2000     

Monte Carlo estimation of the maximal flow distribution with discrete stochastic arc capacity levels

This article describes a highly efficient Monte Carolo sampling plan for estimating ordinates of the complementary distribution function of the maximal s-t flow in a directed network with source node s, terminal node t and whose arcs assume random capacities. Communication, electric power, transportation, and water distribution systems fit this representation. The article derives point estimates and interval estimates that hold for finite sample sizes, and shows the extent to which the proposed technique saves time, as compared to crude Monte Carlo sampling, in achieving a specified accuracy.