Intermittent Demand Forecasting. John E. Boylan

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sigma-summation Subscript j equals 0 Superscript upper R minus 1 Baseline sigma-summation Subscript x Baseline sigma-summation Subscript y Baseline double-struck upper P left-parenthesis upper D Subscript upper L plus j Baseline equals x right-parenthesis double-struck upper P left-parenthesis d Subscript upper L plus j plus 1 Baseline equals y right-parenthesis left-parenthesis y minus left-parenthesis upper S minus x right-parenthesis Superscript plus Baseline right-parenthesis Superscript plus Baseline Over upper R mu EndFraction"/>

      where , denotes possible total demand values over the periods up to , and denotes possible demand values in the next period, .

      The amount subtracted from 1 represents the proportion of demand that is not satisfied. The denominator represents the average demand over the whole review interval (length ), recalling that is the average demand per unit time period.

      In the numerator, each unit of time in the review interval is analysed separately. For the first unit of time, denoted by in Eq. (3.5), backorders are generated if the demand is in excess of the remaining stock, after taking into account the cumulative demand over periods, . The same argument applies for the second unit of time, denoted by : backorders are generated if the demand is in excess of the remaining stock, after taking into account the cumulative demand over periods, . This argument holds for every unit of time in the review interval. In each case, the backorders are weighted by the probability of demand in the unit of time and the probability of cumulative demand in the preceding periods, and are summed appropriately. The results are totalled over the whole review interval to give the expected total backorders.

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