A distribution multi-echelon lot-size model

This paper targets a fundamental, if thus-far neglected, problem in multi-echelon inventory theory: is there an optimal number of echelons for stock storage? It shows that the answer to this question, under certain conditions, is in the affirmative. As modeling framework, an extended lot-size model with no shortages is chosen, the usual array of underlying regularity assumption, but with the added option of storing inventory in several vertical echelons. The trade-off between stocking two different echelons on the inventory ladder is that the higher one goes the lower the carrying charges are but the higher the handling charges, like delivery cost, customer dissatisfaction, etc., are, and these two have to be balanced out. The initial objective is to minimize the total cost per unit time via the determination of the lot-size, or alternatively the cycle’s length, and simultaneously its distribution among the available echelons, or alternatively again the proration of the inventory cycle into sub-periods in which demand is met by different echelons. After establishing this the final objective of finding the number of echelons for which the aforementioned cost is smallest is being taken up. Under the somewhat restrictive supposition of the constancy of the set-up costs a condition for a unique optimal number of stocking echelons is developed. The paper ends with suggestions for more elaborate models, both deterministic and stochastic, for which a search for an optimal number of echelons is warranted.