In plotting the function, we have ignored the purchase cost component cd, which is constant and would just push the graph up a bit. This is not relevant to us, since what we are interested in is finding an order size Q* minimizing total cost. Indeed, since the function goes to infinity for very small and very large order sizes, we would expect that somewhere in between there is an optimal order size, the EOQ. One possible way of finding the EOQ is by trial and error, i.e., by calculating the cost for several input values and spotting the best choice. Unfortunately, this brute-force approach is time-consuming, not quite informative, and not feasible in more complicated cases. We will learn a more straightforward way to spot a minimum-cost or maximum-profit solution, based on the concept of function derivative. For now, we can observe that the minimum cost solution is a point where the tangent line to the function graph is horizontal. This is illustrated in Fig. 2.3(a): Three tangent lines are shown and, indeed, the minimum-cost solution is where the tangent line is horizontal. It seems that, if the tangent line corresponding to an order size is not horizontal, we may find an improvement by moving to the left or to the right. Actually, this reasoning is not 100% correct, and this intuition needs some refinement. Still, using concepts explained later along these lines, it will be easy to see that the optimal order size, according to the EOQ model, is
We see that the EOQ size is increasing with respect to fixed charge A and decreasing with respect to inventory holding cost h, which is quite reasonable.
Fig. 2.3 (a) Tangent lines to the total cost function, and (b) the optimal cost in the EOQ model for varying demand rate d.
If we plug the optimal order size into the total cost function, we get the average cost per year for the optimal solution as a function of the demand rate d:
This function is plotted in 2.3(b), and it is interesting to note its shape. As expected, it is an increasing function of the demand rate. This is no surprise, after all: The larger the demand that must be satisfied, the larger the incurred cost. What is not that obvious is that the rate at which the function increases is decreasing. To see this, imagine drawing tangent lines at different points on that graph. The slopes of the tangent lines are decreasing with respect to demand rate d. This is a typical behavior of cost functions exhibiting economies of scale. We will learn later that a function like that is a concave function.
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