In order to express average total cost per unit time as a function of the order size Q, we should consider all of the factors contributing to the overall cost. The first one that comes to mind is purchase cost. If the unit item cost is c, measured in money per item, we have to pay cQ whenever we replenish; this variable cost per order must be translated in terms of cost per unit time. If we issue an order every Q/d years (our time unit), then we will issue d/Q orders per year, on the average. For instance, if demand rate is 1200 items per year and the order size is Q = 100, we will issue 12 orders per year. Hence, the contribution of purchasing to total cost is just the product of cost per order times average number of orders:
On second thought, this is obvious: In order to satisfy total demand during one year, we have to order d items, anyway, whether in small or large batches. We notice immediately that this term does not depend on Q, as we are disregarding possible discounts based on ordered quantity, which would make the unit cost a function c(Q) of the order size.
Now let us consider the contribution to total cost from the fixed-charge component. Whenever we order, we have to pay A (euros, dollars, or whatever), and we have already found that the average number of orders we issue per year is d/Q, on the average. Then, the contribution from the fixed ordering charge is
Finally, we need the contribution due to inventory holding. This is a bit trickier, and we should clarify which kind of cost we are dealing with. If we keep one item in inventory for one year, we incur a cost. Let us denote this unit inventory holding cost by h. It is important to realize that the dimensions of this unit inventory holding cost are money per item, per unit time; in a sense, this is a “twice” unit cost, with respect to volume and with respect to time. There are a few ways to figure out the contribution from holding cost and all of them require a careful look at Fig. 2.1. There, we see that the inventory level ranges between 0 and Q and changes according to a smooth and constant rate. The pattern is repeated every Q/d time units. Consider an order for Q items. A good question is: How much time does an item of that shipment spend sitting in our warehouse? Actually, the first item that gets issued from inventory waits no time at all, in our abstract model; it is placed into the warehouse when we receive the shipment, but it is immediately sold. The less lucky item is the last one, as it will wait Q/d time units. On average, the waiting time for an item will just be the average between the two limit values 0 and Q/d, i.e., Q/(2d). Since there are d items going through the inventory each year, the contribution to total cost from inventory holding is
As an alternative way to obtain the same result, we may consider the average inventory level. Since the inventory level ranges between Q to 0 and changes uniformly in time, its average is Q/2. Multiplying average inventory level by h yields the result above. This alternative view has the advantage of being generalized to any inventory pattern, not necessarily a constant and uniform one. Now we are ready to put all of it together. The average total cost per year, as a function of Q, is
One thing is immediately clear: The purchase cost component is constant and does not play any role in determining the optimal order size. If you have any difficulty in seeing this, please have another look at Fig. 1.2.
As we see, a function f(·) is essentially a rule mapping an independent variable, Q in our case, into a value y = f(Q) of a dependent variable. Notations like f(·) or f are used to emphasize the difference between the function itself (a rule to map variables to values) and the output value f(Q) taken for an input value Q (a specific numerical value).
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