Name: Student ID: PIN: 1. [100 points + Bonus 10 points] You took a midterm exam of 'Management Science' taught by Prof. Kim and you have just won a $10,000 prize for getting the highest score. You are setting aside $4,000 for taxes and party expenses, and you consider investment of up to $6,000. Upon hearing this news, two friends - HyunJoon and KyungJoon-have offered you an opportunity to become a partner in two different entrepreneurial ventures, one planned by each friend. In both cases, this investment would involve expending some of your time next summer as well as putting up cash. Becoming a full partner in HyunJoon's venture would require an investment of $5,000 and 400 hours, and your estimated profit (ignoring the value of your time) would be $4,000. The corresponding figures for KyungJoon's venture are $ a and 500 hours, with an estimated profit to you of $6,000. However, both friends are flexible and would allow you to come in at any fraction of a full partnership you would like. If you choose a fraction of a full partnership, all the above figures given for a full partnership (money investment, time investment, and your profit) would be multiplied by this same fraction. Because you were looking for an interesting summer job anyway (maximum of 600 hours), you have decided to participate in one or both friends' ventures in whichever combination would maximize your total estimated profit. Since Hyunjoon is your close friend, you decide to participate in his venture at least at / % of a full partnership. In addition, since KyungJoon is your best friend, you decide to participate in his venture at least at 8% of a full partnership. You now need to solve the problem of finding the best combination. [PART 2] (d) [20 points] If your estimated profit for the second venture (cz) decreases from $6,000 to $5,000, what will happen to your original optimal solution? Do you still have a unique optimal solution? If not, provide the set of all optimal solutions by using convex combination. (e) [10 points] If your estimated profit for the second venture (c) decreases from $6,000 to $4,000, does your original optimal solution still remain optimal? If not, find a new optimal solution. (f) [20 points] Find an allowable range for your estimated profit for the second venture (cz) so that your original optimal solution remains optimal. (for example, $2,000 $8,000) (g) [Bonus 10 points] Go back to the original problem. In order to possibly increase your total profit, you may consider investment of up to $7,000 by reducing party expenses. How does this condition change your optimal solution and corresponding total profit? Name: Student ID: PIN: 1. [100 points + Bonus 10 points] You took a midterm exam of 'Management Science' taught by Prof. Kim and you have just won a $10,000 prize for getting the highest score. You are setting aside $4,000 for taxes and party expenses, and you consider investment of up to $6,000. Upon hearing this news, two friends - HyunJoon and KyungJoon-have offered you an opportunity to become a partner in two different entrepreneurial ventures, one planned by each friend. In both cases, this investment would involve expending some of your time next summer as well as putting up cash. Becoming a full partner in HyunJoon's venture would require an investment of $5,000 and 400 hours, and your estimated profit (ignoring the value of your time) would be $4,000. The corresponding figures for KyungJoon's venture are $ a and 500 hours, with an estimated profit to you of $6,000. However, both friends are flexible and would allow you to come in at any fraction of a full partnership you would like. If you choose a fraction of a full partnership, all the above figures given for a full partnership (money investment, time investment, and your profit) would be multiplied by this same fraction. Because you were looking for an interesting summer job anyway (maximum of 600 hours), you have decided to participate in one or both friends' ventures in whichever combination would maximize your total estimated profit. Since Hyunjoon is your close friend, you decide to participate in his venture at least at / % of a full partnership. In addition, since KyungJoon is your best friend, you decide to participate in his venture at least at 8% of a full partnership. You now need to solve the problem of finding the best combination. [PART 2] (d) [20 points] If your estimated profit for the second venture (cz) decreases from $6,000 to $5,000, what will happen to your original optimal solution? Do you still have a unique optimal solution? If not, provide the set of all optimal solutions by using convex combination. (e) [10 points] If your estimated profit for the second venture (c) decreases from $6,000 to $4,000, does your original optimal solution still remain optimal? If not, find a new optimal solution. (f) [20 points] Find an allowable range for your estimated profit for the second venture (cz) so that your original optimal solution remains optimal. (for example, $2,000 $8,000) (g) [Bonus 10 points] Go back to the original problem. In order to possibly increase your total profit, you may consider investment of up to $7,000 by reducing party expenses. How does this condition change your optimal solution and corresponding total profit