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311.
We describe a modification of Brown's fictitious play method for solving matrix (zero-sum two-person) games and apply it to both symmetric and general games. If the original game is not symmetric, the basic idea is to transform the given matrix game into an equivalent symmetric game (a game with a skew-symmetric matrix) and use the solution properties of symmetric games (the game value is zero and both players have the same optimal strategies). The fictitious play method is then applied to the enlarged skew-symmetric matrix with a modification that calls for the periodic restarting of the process. At restart, both players' strategies are made equal based on the following considerations: Select the maximizing or minimizing player's strategy that has a game value closest to zero. We show for both symmetric and general games, and for problems of varying sizes, that the modified fictitious play (MFP) procedure approximates the value of the game and optimal strategies in a greatly reduced number of iterations and in less computational time when compared to Brown's regular fictitious play (RFP) method. For example, for a randomly generated 50% dense skew-symmetric 100 × 100 matrix (symmetric game), with coefficients |aij| ≤ 100, it took RFP 2,652,227 iterations to reach a gap of 0.03118 between the lower and upper bounds for the game value in 70.71 s, whereas it took MFP 50,000 iterations to reach a gap of 0.03116 in 1.70 s. Improved results were also obtained for general games in which the MFP solves a much larger equivalent symmetric game. © 1996 John Wiley & Sons, Inc. 相似文献
312.
M. L. Chaudhry 《海军后勤学研究》1972,19(2):369-378
The technique of probability generating functions has been applied to solve the steady state behavior of a discrete-time, single-channel, queueing problem wherein the arrivals to the queue at consecutive time-marks are statistically independent, but the service is accomplished in phases which are Markov-dependent. Special cases of importance have been discussed. In the end, mean number of phases, its special cases, the mean queue lengths, and the variances have been ascertained. 相似文献
313.
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. 相似文献
314.
F. M. Scherer 《海军后勤学研究》1966,13(1):71-82
This paper explores the relationship between research project cost and expected time to completion under various scheduling strategies; it assumes that many potential technical approaches to the research problem can be identified; and that each approach has a low but finite subjective probability of success. It is shown that under a variety of assumptions, expected time to project completion can be reduced, but that as a result expected project cost rises at an increasing rate. Some cases in which this convex time-cost tradeoff relationship might not hold generally are identified. When the time-cost tradeoff function is convex, the desirability of concurrent as opposed to series scheduling of approaches depends crucially upon the depth of the stream of benefits expected to be realized upon successful project completion. The deeper the benefit stream is, the more desirable concurrent scheduling is. 相似文献
315.
This paper presents a statistical decision analysis of a one-stage linear programming problem with deterministic constraints and stochastic criterion function. Procedures for obtaining numerical results are given which are applicable to any problem having this general form. We begin by stating the statistical decision problems to be considered, and then discuss the expected value of perfect information and the expected value of sample information. In obtaining these quantities, use is made of the distribution of the optimal value of the linear programming problem with stochastic criterion function, and so we discuss Monte Carlo and numerical integration procedures for estimating the mean of this distribution. The case in which the random criterion vector has a multivariate Normal distribution is discussed separately, and more detailed methods are offered. We discuss dual problems, including some relationships of this work with other work in probabilistic linear programming. An example is given in Appendix A showing application of the methods to a sample problem. In Appendix B we consider the accuracy of a procedure for approximating the expected value of information. 相似文献
316.
Carl M. Harris 《海军后勤学研究》1967,14(2):219-230
The purpose of this paper is to explore an extension of the output discipline for the Poisson input, general output, single channel, first-come, first-served queueing system. The service time parameter, μ, is instead considered a random variable, M. In other words, the service time random variable, T, is to be conditioned by a parameter random variable, M. Therefore, if the distribution function of M is denoted by FM(μ) and the known conditional service time distribution as B(t |μ), then the unconditional service distribution is given by B(t) = Pr {T ≤ t}. = ∫-∞∞ B(t |μ) dFM(μ). Results are obtained that characterize queue size and waiting time using the imbedded Markov chain approach. Expressions are derived for the expected queue length and Laplace-Stieltjes transforms of the steady-state waiting time when conditional service times are exponential. More specific results are found for three special distributions of M: (1) uniform on [1.2]; (2) two-point; and (3) gamma. 相似文献
317.
Carl M. Harris 《海军后勤学研究》1967,14(2):231-239
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, nj = Pr { j customers in system at an arbitrary point of departure}, of the imbedded chain, {Xn/Xn = number in system after nth customer is serviced}, is obtained, and the steady-state probabilities, themselves, are found in closed form. 相似文献
318.
Basheer M. Khumawala 《海军后勤学研究》1973,20(1):109-121
This paper introduces an efficient heuristic procedure for a special class of mixed integer programming problems called the uncapacitated warehouse (plant) location problem. This procedure is derived from the branching decision rules proposed for the branch and bound algorithm by the author in an earlier paper. It can be viewed as tracing a single path of the branch and bound tree (from the initial node to the terminal node), the path being determined by the particular branching decision rule used. Unlike branch and bound the computational efficiency of this procedure is substantially less than linearly related to the number of potential warehouse locations (integer variables) in the problem. Its computational efficiency is tested on problems found in the literature. 相似文献
319.
320.