When two competing companies, A and B, plan to advertise their products in a market for the advertising campaign term which can be divided into some periods, they face the problem of allocation of their advertising budgets to each period over the term. In this paper, this problem is treated with the techniques of game theory. The effect of advertising expenditure allocated for a certain period in the advertising campaign term appears not only for this period but also for the periods after it. This after-effect that may be called "carry-over effect" is taken into account in the determination of the optimal allocation of the advertising budget. The following assumptions is made. (1) The advertising campaign term can be divided into n periods. (2) Companies A and B have fixed advertising budgets A and B respectively. (3) The potential sales in the ith period is s_i, where Σ^<i=n>_<i=1>=S, the total potential sales. (4) The carry-over effect of the advertising expenditure of a allocated for a certain period corresponds to the advertising expenditure of ar^k (0≦r≦1) after k (k=1,2,...) periods. (5) Sales by each company for the ith period can be calculate by multiplying s_i by his share of cumulative advertising capital (the sum total of the advertising expenditure for this period and the carry-over effect of all past advertising expenditures). On these assumptions, let the pland of the advertising budget allocation of companies A and B for the advertising campaign term be A=(a_1,a_2...,a_n) and B=(b_1, b_2...,b_n) respectively, and difference M(A, B) in total sales between companies A and B will be given by the equation [numerical formula] If this difference is taken as payoff to the company A and negative as payoff to the company B, this problem is treated as a two-person zero-sum game in which the company A tries to maximize M(A, B) and the company B to minimize it.

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Maximum likelihood estimates of parameters in continuous time Markov chains are obtained when the observation plan is Poisson sampling. Furthermore, for birth and death processes, variance-covariance matrix of parameters is obtained. Especially, for queueing model M/M/1, the variance-covariance matrix by Poisson sampling is compared with the variance-covariance matrix by complete observation in detail.

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An approximation method based on the diffusion theory is proposed for solving multi-server finite queueing problems having general independent inter-arrival time and general service time distributions. The discrete customer flow through the system is approximated by a continuous one, and a diffusion equation for the process of the number of customers in the system is constructed by using means and variances of the inter-arrival time and service time distributions. Two reflecting boundaries are imposed at the origin and m, the maximum number of customers being allowed in the system. Later, the boundary conditions are modified to improve the approximation. Approximate formulas for P_n. Probability of finding n customers in the system, and for N, mean number of lost customers from the system per mean service time, are given for steady state. Numerical examples for mean number of customers in the system are presented for some E_l/E_k/s(m) systems to show the effectiveness of the proposed method.

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A two-person zero-sum time sequential game related to the sequential stochastic assignment problem is considered. The i.i.d. random variables appear one by one at a time. At each stage, after observing the value x of the random variable, both players must decide whether to accept or reject, and if player I who has n men to assign decides toaccept, then he selects one of the n men in order to assign. When both players accept, this random variable is selected and assigned to the selected man with the ability p. The immediate payoff to player I is px. After the man is assigned, he is unavailable for the future stages. If otherwise, they get a new observation and t,he selected man is assigned in the future stages. The objective of player I (resp. player II) is to maximize (resp. minimize) the total expected reward. In this paper we obtain therecursive equation from which we get the optimal strategies of both players and the value of the game. The optimal strategies of the players are similar to that of the stochastic assignment problem. We observe the property of the special case of this game in the asymptotic case.

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This paper considers the stability of the waiting time in the queueing system with the critical condition: ES_i = sET_i. It is shown that the waiting time is stable in the model with special moving average input and the unstability condition is obtained in the strictly stationary input with the finite variance.

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A simple-and-general closed form of the long-run distribution of stationary inventory position processes in < R, r, T > inventory systems is formulated in recurrence relations, the computational procedure of which is also applicable to determine the long-run distribution of non-stationary inventory position processes in those systems as long as their limits exist. Moreover, its computation on computer is economically plausible so that a significant contribution to the analytical study on such inventory problems is greatly anticipated.

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Fuzziness must be considered in systems where human estimation is influential. Even if inputs are not fuzzy, outputs may become fuzzy because of fuzziness of system equation. In the background of usual regression model, deviations between the observed values and the estimated values are supposed to be due to the measurement errors. On the contrary it is assumed in our paper that these deviations depend on indefiniteness of system structure, which can be grasped by the concepts of possibility, that is fuzzy sets. We regard these deviations as the fuzziness of parameters of system. Thus these deviation is reflected in a fuzzy linear function which is used as a linear model in a fuzzy environment. As an example of this problem, we have got the fuzzy linear model which explains a price mechanism of prefabricated house. The fuzzy parameter of linear model obtained by our method means a possibility distribution, which corresponds to fuzziness of the system. This fuzzy regression model might be very useful for finding out a fuzzy system structure like an evaluation system.

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One stationary object is in one of n boxes with the distribution <p_1,…p_n>. Let F_i(t) be the distribution of the lifetime of the object in box i and suppose that F_i(t) is composed of two probability masses α_i at t = 0, β_i at t = ∞ and a probability density function f_i(t) on the interval (0, ∞) which is differentiable in t almost everywhere. Let c be the search cost per unit time. If the object is in box i and box i is searched for t hours, the object is detected with probability 1-exp(-λ_it) whether it is alive or not. We suppose that the search is continued until the object is detected whether it is alive or not. If the searcher detects the living object in box i, he obtains a reward r_i(>0). If the searcher detects the died object, no reward is obtained. The criterion is to maximize the expected return (reward minus cost) until the object is detected. We obtain necessary and sufficient conditions for a policy to be optimal. Furthermore we obtain the optimal search rate in the implicit form. Specially we obtain the optimal search policy. explicitly in the case that f_i(t) (i = 1,…, n) are differentiable in t. We consider two numerical examples and give the explicit solutions. One of them is the case of the exponential lifetime distribution and another is the case of the uniform lifetime distribution. Finally we deal with stopping problem in which the searcher is permitted to stop the. search at any time. Some results about the optimal search policy and the optimal stopping time are obtained.

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We consider an n-job one machine scheduling problem in which the processing time of each job i is a random variable subject to a normal distribution N(m_i, v^2_i) and the object is to maximize the weighted number of early jobs subject to the constraint that some specified jobs must be early. It is assumed that m_p &it; m_q implies v^2_p≦v^2_q, where m_i and v^2_i are, respectively, a known mean value and a known variance associated with each job i. If such constraint is relaxed, the problem has been shown to be NP-complete, suggesting strongly that there exists no efficient exact algorithm whatever for the problem. Moreover, It is assumed that if m_p < m_q or v^2_p < v^2_q, then w_p ≧ w_q, where w_i is a known weight associated with each job i. It is well known for the problem with arbitrary weights to be NP-complete even in the deterministic case (i.e., v^2_i = 0). We show that the problem with the above assumptions can be solved in O(n^2) time and that it has a practical application.

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