When a careful count was taken a few years back, the Timbuktu Post Office handled at least 3120 letters and 1890 advertising brochures daily. The canceling machine then in use could handle 390 letters and 315 brochures per hour at an hourly cost of $2.10. The post office was planning to acquire a new and improved model which could handle 520 letters and 210 brochures per hour at a cost of $2.70 per hour. Neither machine could be run for more than 10 hours per day due to internal heating. Neither machine could handle more than the maximum per hour in either category. For example, if the current machine was canceling only brochures, it still couldn’t handle more than 315 brochures per hour, and it still couldn’t run more than 10 hours per day without melting down. With these constraints, what would be the most economical way to run the post office canceling operation if they purchased the new machine?
Well, the old machine can handle the workload of 3120 letters and 1890 brochures by running 8 hours per day at a cost of $16.80 per day.
We note that the old and new machines cost the same for each letter cancelled ($.005 per letter) but the old machine is cheaper for the brochures ($.007 per brochure vs. $.013 per brochure for the new machine).
If we run the old machine long enough to cancel all the brochures (6 hours), then we have 3120 – 6(390) = 780 letters still to cancel. Since the new machine works faster on letters, it takes the new machine only 1.5 hours to cancel the remaining letters.
Thus the total cost is 6($2.10) + 1.5($2.70) = $16.65, which is $.15 less than the old machine working by itself.
Is it worth investing in a new machine to save as little as 15 cents per day? If they decide to go ahead and get the new machine with an eye toward handling a larger volume of mail, the most economical processing will occur if the old machine is used, up to capacity, to cancel brochures and the new machine is used to cancel the remaining letters, if any.