Nonstationary stochastic gold-mining: A time-sequential tactical-allocation problem

This paper presents an extension of gold-mining problems formulated in earlier work by R. Bellman and J. Kadane. Bellman assumes there are two gold mines labeled A and B, respectively, each with a known initial amount of gold. There is one delicate gold-mining machine which can be used to excavate one mine per day. Associated with mine A is a known constant return rate and a known constant probability of breakdown. There is also a return rate and probability of breakdown for mine B. Bellman solves the problem of finding a sequential decision procedure to maximize the expected amount of gold obtained before breakdown of the machine. Kadane extends the problem by assuming that there are several mines and that there are sequences of constants such that the jth constant for each mine represents the return rate for the jth excavation of that mine. He also assumes that the probability of breakdown during the jth excavation of a mine depends on j. We extend these results by assuming that the return rates are random variables with known joint distribution and by allowing the probability of breakdown to be a function of previous observations on the return rates. We show that under certain regularity conditions on the joint distributions of the random variables, the optimal policy is: at each stage always select a mine which has maximal conditional expected return per unit risk. This gold-mining problem is also a formulation of the problem of time-sequential tactical allocation of bombers to targets. Several examples illustrating these results are presented.     

The search for an intelligent evader: Strategies for searcher and evader in the two-region problem

This paper considers the search for an evader concealed in one of two regions, each of which is characterized by its detection probability. The single-sided problem, in which the searcher is told the probability of the evader being located in a particular region, has been examined previously. We shall be concerned with the double-sided problem in which the evader chooses this probability secretly, although he may not subsequently move: his optimal strategy consists of that probability distribution which maximizes the expected time to detection, while the searcher's optimal strategy is the sequence of searches which limits the evader to this expected time. It transpires for this problem that optimal strategies for both searcher and evader may generally be obtained to a surprisingly good degree of approximation by using the optimal strategies for the closely related (but far more easily solved) problem in which the evader is completely free to move between searches.     

A note on the limitations of goal programming as observed in resource allocation for marine environmental protection

After first formulating the problem of the Marine Environmental Protection program of the Coast Guard as a multiple-objective linear program, we investigate the applicability and limitations of goal programming. We point out how the preemptive goal-programming approach is incompatible with utility preferences. Then we observe the tendency of optimal solutions for standard linear goal programs to occur at extreme points. We also note problems of more general approaches, such as dealing with additively separable approximations to preferences.     

A heuristic network procedure for the assembly line balancing problem

Proposed is a Heuristic Network (HN) Procedure for balancing assembly lines. The procedure uses simple heuristic rules to generate a network which is then traversed using a shortest route algorithm to obtain a heuristic solution. The advantages of the HN Procedure are: a) it generally yields better solutions than those obtained by application of the heuristics, and b) sensitivity analysis with different values of cycle time is possible without having to regenerate the network. The rationale for its effectiveness and its application to problems with paralleling are presented. Computational experience with the procedure on up to 50 task test problems is provided.     

A heuristic for maximizing the number of on‐time jobs on two uniform parallel machines

We consider the problem of maximizing the number of on‐time jobs on two uniform parallel machines. We show that a straightforward extension of an algorithm developed for the simpler two identical parallel machines problem yields a heuristic with a worst‐case ratio bound of at least . We then show that the infusion of a “look ahead” feature into the aforementioned algorithm results in a heuristic with the tight worst‐case ratio bound of , which, to our knowledge, is the tightest worst‐case ratio bound available for the problem. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2006     

An approximative algorithm for the fixed charge problem

This paper describes an approximate solution method for solving the fixed charge problem. This heuristic approach is applied to a set of test problems to explore the margin of error. The results indicate that the proposed fixed charge simplex algorithm is capable of finding optimal or near optimal solutions to moderate sized fixed charge problems. In the absence of an exact method, this heuristic should prove useful in solving this fundamental nonlinear programming problem.     

Confidence intervals for ranked means

Suppose that observations from populations π₁, …, π_k (k ≥ 1) are normally distributed with unknown means μ₁., μ_k, respectively, and a common known variance σ². Let μ_[1] μ … ≤ μ_[k] denote the ranked means. We take n independent observations from each population, denote the sample mean of the n observation from π₁ by X _i (i = 1, …, k), and define the ranked sample means X _[1] ≤ … ≤ X _[k]. The problem of confidence interval estimation of μ₍₁₎, …,μ_[k] is stated and related to previous work (Section 1). The following results are obtained (Section 2). For i = 1, …, k and any γ(0 < γ < 1) an upper confidence interval for μ_[i] with minimal probability of coverage γ is (? ∞, X _[i]+ h) with h = (σ/n^1/2) Φ^?1(γ^1/k-i+1), where Φ(·) is the standard normal cdf. A lower confidence interval for μ_[i] with minimal probability of coverage γ is (X _i[i] – g, + ∞) with g = (σ/n^1/2) Φ^?1(γ^1/i). For the upper confidence interval on μ_[i] the maximal probability of coverage is 1– [1 – γ^1/k-i+1]ⁱ, while for the lower confidence interval on μ_[i] the maximal probability of coverage is 1–[1– γ^1/i] ^k-i+¹. Thus the maximal overprotection can always be calculated. The overprotection is tabled for k = 2, 3. These results extend to certain translation parameter families. It is proven that, under a bounded completeness condition, a monotone upper confidence interval h(X ₁, …, X _k) for μ_[i] with probability of coverage γ(0 < γ < 1) for all μ = (μ_[1], …,μ_[k]), does not exist.     

A method for solving the transportation problem

A new method has been developed f o r solving the transportation problem. This method is a modification and a generalization of the method for solving the multiple assignment problem developed by Dr. A. J. Hoffman and Dr. H. M. Markowitz. This method is of interest for several reasons. In the first place it is significantly different from all other methods for solving the trans -portation problem known to the author. Secondly, it is moderately simple touse and understand. Thirdly, and perhaps most important, it has proved to be very adaptable tri high-speed computer operations. It is now being used by several branches of the armed services.