Suppose that chemical X is manufactured using a raw material B that is available from a location called the "mine." Production of one ton of X requires 1/3 of a ton of B. A firm called X ENTERPRISES, which has a contract to deliver 30 tons of X to a location called the "market," is trying to decide where to locate its plant. The mine and the market are 50 miles apart. Overland shipping of both X and B costs $2 dollars per ton per mile shipped. However, additional costs must be incurred because a river passes between the mine and the market, and the river has no bridge. Goods must be loaded onto barges to cross the river, which is located 16 miles from the mine. Barge operators charge $1 dollar per ton of X shipped across the river. However, since the input B is highly toxic when mixed with water, barge operators must charge an extremely high price to transport B across the river. This price defrays the cost of insurance that the operators must carry to meet liability claims should they accidentally pollute the river with their cargo. The cost of shipping one ton of B across the river is $195.
A) Using the above information, find the transport-cost-minimizing location for X ENTERPRISES. The answer can be found by computing transport costs at four locations: mine, market, mine side of the river, and market side of river. Show your work. You can assume that the width of the river is negligible, so that it can be ignored.
B) Illustrate your results in a carefully-drawn diagram like that presented in Figure 1.6 (use graph paper, if preferred). Graph the input shipping-cost curve by plotting the input shipping costs at the same four locations as in part (a) and then connecting the dots (the curve is drawn backwards). Similarly, plot the output shipping cost at the four locations, and then connect the dots to generate the output shipping-cost curve. Then, graph the total shipping-cost curve by adding the input and output shipping costs at each of the four locations, plotting the points and connecting the dots. Using the diagram, identifying the best location for X ENTERPRISES, which should be the same as your answer in part (a). Note that the shipping-cost curves for this problem are straight lines with jumps at the river
C) Give an intuitive explanation of your results.
D) Suppose that a bridge were built across the river, which would eliminate the cost of crossing it. Redo parts (a), (b), and (c) under this assumption.
