This is an "Economic Order Quantity" question. The first step is to determine the quantity to order and how many times per year to order.
EOQ = sqrt(2 * Annual Demand * Cost per Order / Annual Holding Cost per Unit)
Then, plug in the values given in the question:
EOQ = sqrt(2*8,000*50 / (20%*19)) = 458.83 units per order (note that since this is above the min of 440, then we know this is a potential solution).
Now we need to know how many times per year to order...8,000/458.83 = 17.6 times. Of course you can't order .6 times so we'll round to either 17 or 18. If it's 17 then you need to order 471 units per order (i.e., 8,000/17)...if it's 18 then you need to order 445 units per order (i.e., 8,000/18).
Now we'll need to compare Total Annual Costs for each solution:
If we order 445 units 18 times per year: TAC = 18*50+18*19*445+445/2*20%*19 = $153,935.50
If we order 471 units 17 times per year: TAC = 17*50+17*19*471+471/2*20%*19 = $153,877.90
Since the latter is cheaper, we'd select this option...note, however, that on the last order of the year we could actually elect to order only 464 units to get to exactly 8,000 units for the year, thus we could reduce the cost by 7*19 = 133...thus the total annual cost would fall to 153,877.90 - 133 = 153,744.90.
As noted in the original problem there are a couple rounding issues so we'll choose the closest answer which is $153,743 (only $1.90 away from our answer)
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