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A new business graduate has been given the task of determining the best production quantity for a new product that will be produced on a new...

The Economic Order Quantity (EOQ) is the optimal solution for how much product to order, assuming that you are buying from a vendor and all orders will be filled completely and immediately. For producing a product, the optimal solution would actually be the Economic Production Quantity (EPQ), but in this case we couldn't determine that because we didn't know the production rate.

For an order/setup cost of K, a demand rate of D, a holding cost of F, and a production rate of P, these are the equations for each quantity:

EOQ = sqrt(2DK/F)

EPQ = sqrt(2DK/(F(1-D/P))

Thus, the two quantities differ only by a ratio of 1/sqrt(1-D/P). We are given that the demand rate is 80% of the production rate, so D/P = 0.8, and 1/sqrt(1-D/P) = 2.24.

Therefore, we are not holding enough of the product, by a factor of 2.24. In other words, we are only holding 44.7% of the quantity we should be. This is because the production rate is only slightly faster than the demand rate, so we can barely keep up with demand if we don't hold onto inventory and thus end up incurring a lot of per-run costs by doing so many different production runs.

The sensitivity analysis formula says that the ratio in average cost between the actual production and the optimal production is, for actual quantity Q and optimal quantity Q*, the average of their ratios:

AC(Q)/AC(Q*) = 1/2 * (Q/Q* + Q*/Q)

Since Q/Q* = 2.24 and Q*/Q = 0.447, we just substitute in:

AC(Q)/AC(Q*) = 1/2 (2.24 + 0.447) = 1.3435

So, we are paying 34% more than we should be.

Still, I think the consultant can make a good case that the EOQ was a reasonable choice; it would be the best choice if production rate is much higher than demand rate, and even in this particular case it was not wildly wrong. The employer, on the other hand, might argue that they should have made some ballpark assumption for the production rate and gone with that, say assume that D/P = 0.5; because effectively using the EOQ assumes D/P = 0, which is obviously not realistic.

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