The Determination of the Optimum Joint Replenishment Policy for a Group of Discount-Connected Stock Lines

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Abstract
This paper investigates the situation where a joint replenishment order for a group of stock lines from a common supplier qualifies for a quantity discount provided the total order equals a specified number of units. It is assumed that, when any line in the group reaches its reorder point, an order of this size is made for the whole group and it is the problem of how to determine the optimum order "mix" which is considered here. In the case of a group consisting of two lines only, an analytical solution is derived which, assuming Poisson sales-rates, gives the individual stock levels for which the expected number of sales between successive replenishments is maximized for a given total stock. However, for groups of more than two lines, it was necessary to resort to numerical methods and a computer program is described which uses a quasi dynamic-programming approach to determine the optimum set of individual stocks for any given set of sales-rates and total stock. Based on the results from this program, a "ready reckoner" has been constructed which permits an easy look-up to obtain the optimum allocation of any given total stock between the lines of a group of any size and sales-rates. Finally, the possible extension of this approach to other multi-item inventory problems is discussed.

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