Maintaining optimal inventory level by feedbacks

In this paper the inventory level is examined under a situation in which the demand function consists of two parts. One, which is sometimes referred to as the trend of demand, is described as a proper polynomial function of time. The other is given as a random variable with a finite variance. A policy is planned which is to bring the expected value of realized inventory level to the planned level. In order to accomplish this purpose, some feedbacks are to be introduced. The first feedback is realized by estimating the demand level in the future based upon the demand level in the past. How many data are necessary depends on the degree of the polynomial function to be estimated. The second is obtained by monitoring the realized inventory level. The difference between the realized inventory level and the planned level is used as a feedback. We will show that by combining these two feedbacks, the expected value of difference between two inventory levels, planned and realized, converges to zero.