Alex is a high school student who scored 75 out of 100 points in the math...

 

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Alex is a high school student who scored 75 out of 100 points in the math midterm exam, 20 out of 100 in the Spanish midterm exam, and 65 out of 100 in the physics midterm exam. Alex has the option to hire private tutors to improve his exam scores. The table below estimates the score Alex will get in the final exams after attending a certain number of hours of tutor sessions and the rate of tutors for the three subjects respectively. The highest score one can get for each subject is 100 points. We assume that Alex will get the same grade in the final exams, i.e., 75 for math, 20 for Spanish and 65 for physics, if he does not hire tutors at all. Alex is able to spend at most 40 hours on tutoring sessions before the final exam. Part (a) Formulate the optimization problem (6 pts) Alex wants to maximize the total score of the three subjects in the final exams and minimize the tuition fees (cost of hiring private tutors). Formulate the problem as a single-objective optimization problem. Your formulation should clearly specify: 1. Decision variables 2. Objective function 3. Constraints Your answer here: Part (b) Re-formulate the optimization problem (10 pts) Now suppose that Alex wants to maximize the total score of the three subjects and is no longer concerned by tuition fees. Reformulate the problem and approximate the problem by linearizing the problem using piece-wise linear functions. Your formulation should clearly specify: 1. Decision variables 2. Objective function 3. Constraints Number of hours Final exam score Rate ($) Math xm 75 + Xm 50xm Physics Xp x +6x + 20 85 - 20e S 10 S 60xp Spanish X s 40% s xm +xp 20 Alex is a high school student who scored 75 out of 100 points in the math midterm exam, 20 out of 100 in the Spanish midterm exam, and 65 out of 100 in the physics midterm exam. Alex has the option to hire private tutors to improve his exam scores. The table below estimates the score Alex will get in the final exams after attending a certain number of hours of tutor sessions and the rate of tutors for the three subjects respectively. The highest score one can get for each subject is 100 points. We assume that Alex will get the same grade in the final exams, i.e., 75 for math, 20 for Spanish and 65 for physics, if he does not hire tutors at all. Alex is able to spend at most 40 hours on tutoring sessions before the final exam. Part (a) Formulate the optimization problem (6 pts) Alex wants to maximize the total score of the three subjects in the final exams and minimize the tuition fees (cost of hiring private tutors). Formulate the problem as a single-objective optimization problem. Your formulation should clearly specify: 1. Decision variables 2. Objective function 3. Constraints Your answer here: Part (b) Re-formulate the optimization problem (10 pts) Now suppose that Alex wants to maximize the total score of the three subjects and is no longer concerned by tuition fees. Reformulate the problem and approximate the problem by linearizing the problem using piece-wise linear functions. Your formulation should clearly specify: 1. Decision variables 2. Objective function 3. Constraints Number of hours Final exam score Rate ($) Math xm 75 + Xm 50xm Physics Xp x +6x + 20 85 - 20e S 10 S 60xp Spanish X s 40% s xm +xp 20