An inventory management problem is theoretically discussed for a factory having effects of lead times in replenishing the inventory, where it stocks materials used for its products. It is assumed that the factory can dynamically control the size of ordering materials. By applying the stochastic control theory, the optimal control of the ordering size is derived, in which the expected total cost up to an expiration time is minimized. First, a new stochastic model is constructed for describing an inventory fluctuation of the factory by the use of a non-diffusive stochastic differential equation, where an analytic time is introduced so that the inventory process can be a Markov process even though it is affected by lead times. Next, an optimal control is formulated by introducing an evaluation function quantifying total costs. Based upon them, the Hamilton-Jacobi-Bellman (HJB) equation is derived, whose solution gives the optimal control. Finally, the optimal control is quantitatively examined through numerical solutions of the HJB equation. Numerical results indicate that if time up to an expiration time is short then the optimal control is affected by it, otherwise, the optimal control does not depend on it.
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Statistical estimations for probability distributions having tails of special shape, such as a double-mode distribution as well as the so-called heavy-tailed or fat-tailed distribution, are quantitatively discussed through virtual experiments using computer simulations. In this paper, a probability distribution describing the damage degree of concrete liners of tunnels in cold region is examined as an example of the double-mode probability distribution.First, virtual data sets of observations are generated by the use of quasi-random numbers for a Pareto distribution as a typical example of the fat-tailed distribution, whereas the actual data set obtained for the damage degree of concrete liners is used for generating virtual data set for the double-mode distribution. Next, statistical estimations are executed by the use of probability papers to identify the probability distribution showing the "best" fitting among supposed plural candidates for probability distributions. Finally, the accuracy of the estimation is quantified by applying the coefficient of determination. The results show that the accuracy of the estimation in the tail region is scarcely improved even if the number of data is increased.
A probabilistic model is newly proposed for describing random deterioration of pumping wells, in consideration of application to efficient management of pumping wells using a concept of asset management procedure. First, introducing a health index for quantifying specific capacity, we formulate a differential equation describing mean deterioration of the helth index. Next, it is extended to a random differential equation, in which a Gaussian white noise is introduced for representing intensive fluctuation around the mean deterioration behavior, so that intensively random variation of the health index can be well reproduced. A probability distribution of the health index is then derived in an analytical form. Finally, the obtained probability distribution is compared with actual data of the health index for active eight pumping wells in Joyo area, Kyoto prefecture, Japan. It is clarified that the scatter of health index of actual pumping wells is quite well reproduced by the theoretically derived probability distribution from the newly proposed probabilistic model.
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