Order Size in Inventory Management

by Perry Tsai

Submitted : Fall 2008

This problem concerns with managing the inventory of a single product at a minimum total cost per unit time. It is assumed that: 1. The demand rate R (unit/year) is constant and known. 2. If an order of size Q (> 0) is purchased, it costs K + cQ (where K is a fixed ordering cost and c is the unit price). 3. There is an inventory caring cost of $h/unit/time. That is, if we keep 1 unit of the product in inventory for 1 unit time, a cost of $h incurs. Perry approaches this problem thinking in this way: an order of exactly Q units of the product is placed at equal interval of exactly T time units. So what Perry is doing is to make Q large and if Q is large then we will order less frequently. This may save some ordering costs. But this can lead to large inventory of the product due to T = R/Q, and therefore, large inventory caring costs. On the other hand, if Q is very small, the inventory may be low and the inventory holding cost will also be small. However, Perry will need to order more frequently. This may lead to a large ordering cost. Thus, the objective is to find the optimal order size Q*, what minimizes the total of or ordering and inventory holding cost. So to find the optimum order size Q*, Perry applies calculus to it: AC = Rc + (R/Q)K + (Q/2)I (where AC = annual cost, R = demand rate, c = unit price, Q = quantity, K = fixed ordering cost, and I = inventory caring cost). Perry takes that equation and derives it with respect to Q and then end result of the new equation is: Q* = v[(2RK)/I] (where Q* = optimum quantity, R = demand rate, K = fixed ordering cost, and I = inventory caring cost). Using this equation, Perry was able to draw out the Q* in the few examples in the Discussion portion of the paper.

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