6. ABC Farm is a cranberry grower. It is now April 1st and ABC is about to enter into a contract to sell all its cranberries to the Ocean Place Company. ABC's cranberry crop will be harvested and processed in the fall. The quantity of cranberries produced by ABC is uncertain and follows a normal distribution. ABC estimates that the yield (the quantity of berries) available in the fall has a mean of 10,000 barrels and a standard deviation of 1,500 barrels. For the current contract, ABC must specify now how many barrels it promises to sell to Ocean Place in the fall. The selling price is $500 per barrel, and under the contract, ABC is prohibited from selling cranberries to anyone but Ocean Place. ABC will pay a penalty of $200 for each barrel that is short of what it promises to deliver. For example, if ABC promises to deliver 5,000 barrels but can only deliver 4,500 barrels, Ocean Place would pay ABC a total of 500X4,500200x(5,0004,500). If ABC can deliver all the 5,000 barrels as promised, Ocean Place would pay ABC a total of 500x5,000. On September 1st ABC will know how many berries are available. If the number of barrels is less than the number they have promised to deliver, ABC will process and deliver all of the berries that are available. If the number of barrels available is greater than the number they have promised to deliver, they will process and deliver only the number that has been promised. For each barrel delivered to Ocean Place, ABC also needs to incur a cost of $100 for processing. (a) How much should ABC promise to sell and deliver to Ocean Place? (5 points) (b) With your answer to (a), what is the probability that ABC can NOT deliver the full amount it promises? (5 points) (c) Suppose when ABC cannot harvest and process up to the amount they promise to deliver, they can make up for the shortfall by buying from another grower at a price of $600. No processing cost is charged for this amount. How much should ABC promise to sell and deliver? (5 points) (d) Suppose there is another identical grower, DEF. The quantity of cranberries available to DEF in the fall has a normal distribution with a mean of 10,000 barrels and a standard deviation of 1,500 barrels, and the cost to DEF to process and deliver is $100 per barrel. Assume that buying from another grower to make up for the shortfall is not possible. Suppose DEF can make an agreement with Ocean Place with exactly the same terms as stated above. Therefore, both ABC and DEF would promise to sell and deliver the same amounts of cranberry as your answer to (a). Now suppose ABC merges with DEF. Assume the amounts of cranberry available in the fall to these two growers are independent. How much should the merged company promise to sell and deliver? (5 points) 6. ABC Farm is a cranberry grower. It is now April 1st and ABC is about to enter into a contract to sell all its cranberries to the Ocean Place Company. ABC's cranberry crop will be harvested and processed in the fall. The quantity of cranberries produced by ABC is uncertain and follows a normal distribution. ABC estimates that the yield (the quantity of berries) available in the fall has a mean of 10,000 barrels and a standard deviation of 1,500 barrels. For the current contract, ABC must specify now how many barrels it promises to sell to Ocean Place in the fall. The selling price is $500 per barrel, and under the contract, ABC is prohibited from selling cranberries to anyone but Ocean Place. ABC will pay a penalty of $200 for each barrel that is short of what it promises to deliver. For example, if ABC promises to deliver 5,000 barrels but can only deliver 4,500 barrels, Ocean Place would pay ABC a total of 500X4,500200x(5,0004,500). If ABC can deliver all the 5,000 barrels as promised, Ocean Place would pay ABC a total of 500x5,000. On September 1st ABC will know how many berries are available. If the number of barrels is less than the number they have promised to deliver, ABC will process and deliver all of the berries that are available. If the number of barrels available is greater than the number they have promised to deliver, they will process and deliver only the number that has been promised. For each barrel delivered to Ocean Place, ABC also needs to incur a cost of $100 for processing. (a) How much should ABC promise to sell and deliver to Ocean Place? (5 points) (b) With your answer to (a), what is the probability that ABC can NOT deliver the full amount it promises? (5 points) (c) Suppose when ABC cannot harvest and process up to the amount they promise to deliver, they can make up for the shortfall by buying from another grower at a price of $600. No processing cost is charged for this amount. How much should ABC promise to sell and deliver? (5 points) (d) Suppose there is another identical grower, DEF. The quantity of cranberries available to DEF in the fall has a normal distribution with a mean of 10,000 barrels and a standard deviation of 1,500 barrels, and the cost to DEF to process and deliver is $100 per barrel. Assume that buying from another grower to make up for the shortfall is not possible. Suppose DEF can make an agreement with Ocean Place with exactly the same terms as stated above. Therefore, both ABC and DEF would promise to sell and deliver the same amounts of cranberry as your answer to (a). Now suppose ABC merges with DEF. Assume the amounts of cranberry available in the fall to these two growers are independent. How much should the merged company promise to sell and deliver? (5 points)
