Task 2: plotting the total cost function

Having figured out a relationship between the order size and the average total cost per year, it would be useful to plot the function in order to see the effect of Q and to figure out a good decision. There are plenty of powerful software packages that, given a range of the independent variable Q, compute the corresponding values f(Q) and display the result graphically. This is certainly quite useful, but we should also be able to figure out some basic properties of the function by just looking at its expression. There are a few reasons justifying the need for this ability:

• It is helpful when evaluating the function numerically is difficult or prone to numerical errors.
• It is often the case that a function cannot be evaluated for some values of the independent variable, but a piece of software will not help you in understanding what is going wrong.
• In more complicated settings, we may not just rely on brute computational force; to see this, imagine plotting a function depending on more than two independent variables.
• Last but not least, this exercise helps in honing skills and improving understanding.

Fig. 2.2 Inventory holding and fixed ordering charge components in the EOQ model.

In general, plotting a function can be difficult, but it is fairly easy in the case of the EOQ cost model. Disregarding the constant term, the essential form of the total cost function is

where a = h/2 and b = Ad. From our high school background, we should immediately see that this function is just the sum of a straight line aQ, going through the origin, and a hyperbola b/Q (see Fig. 2.2). To get a rough picture of the function, let us consider the two extreme cases:

• When Q is very small, the linear component aQ is small, too, but the nonlinear component b/Q gets larger and larger; in fact, for a very small order size, the inventory holding cost component goes to zero, but the fixed-charge component increases without limits, because we are spreading the fixed ordering charge over a tiny number of items. We cannot evaluate C_tot(Q) for Q = 0, since we cannot divide by zero, but we can writeThis notation has a precise mathematical meaning, but the informal interpretation is that the limit of the total cost goes to (plus) infinity, when Q tends to zero. Notations likeare used to indicate that Q goes to zero while staying on the positive side. (A negative order size makes no sense!) We say that Q goes to zero “from the right” or “from above.” So, in the range of small order sizes, the nonlinear component prevails and the function looks like a hyperbola.
• When Q is very large, the linear component aQ is large, too, as large holding costs are incurred. On the other side of the coin, the nonlinear component b/Q goes to zero, since the fixed charge component is spread on a huge amount of items. In this range of large order sizes it is the linear component that prevails, and the function looks like a straight line. In this case too, the limit of the cost function goes to infinity:

The result of these observations is a sketch like that in Fig. 2.2. Indeed, we see that there are very bad decisions, corresponding to very large or very small order sizes, and some good ones, for which cost is much lower. Now we have to find the optimal compromise.