only problem 3. I have done the problem but I have doubts concerning my answer

Problem 1: Consider the problem of assigning students to seminars to minimize the average assigned ranking for all students. Let us define the following notation: INPUTS and SETS J set of students K set of seminars rjk ranking of seminar kK by student jJ. A ranking of 1 is the student's first choice, a ranking of 2 is her second choice, and so on. sk Capacity of seminar kK. This is the maximum number of students that can be assigned to the seminar. We need to decide which students should be assigned to which seminars. Thus, we have the following decision variable DECISION VARIABLES Xjk1 if student jJ is assigned to seminar kK;0 if not a) Using this notation, formulate the objective of minimizing the sum or the assigned rankings. Note that this will also minimize the average assigned ranking b) Using this notation, write down the constraints that say that the maximum number of students that can be assigned to a seminar is sk c) Write down a constraint that says that each student is assigned to at least two seminars. d) Write down a constraint that says that for any given seminar a student can be assigned to the seminar at most once. e) Write down the non-negativity constraints. Problem 3: Now let us formulate the dual of the seminar assignment problem. a) Convert the formulation of problem 1 to standard form (a maximization problem with less than or equal to constraints and non-negative variables) b) With this notation, formulate the dual of the seminar assignment problem. Let k be the dual variable associated with the kth seminar capacity constraint of the problem in standard form. Similarly, let j be the dual variable associated with the jth primal student assignment constraint of the problem in standard form, meaning the constraint that says that each student must be assigned to at least two seminars. Finally, let jk be the dual variable associated with the (j,k) constraint stating that any student can be assigned to a seminar at most one time. a. What is the dual objective function? b. What do the dual constraints look like? Write them out using this notation. c) If you did part (a) of problem 2 correctly, only 7 students were assigned to the Math seminar. Given that this is the case, what is the value of the dual variable associated with this seminar? Briefly justify your answer. Note that even if you did not get parts (a) and (b) of this problem done correctly, you should be able to do this part of the problem. d) Based on the Shadow Prices in the Sensitivity Report for Problem 2, part (a), which seminar(s) should you increase to a capacity of 10 if you want to have the most impact on the average assigned ranking? You should verify this answer by resolving the problem for any case you think may have a big impact. Problem 1: Consider the problem of assigning students to seminars to minimize the average assigned ranking for all students. Let us define the following notation: INPUTS and SETS J set of students K set of seminars rjk ranking of seminar kK by student jJ. A ranking of 1 is the student's first choice, a ranking of 2 is her second choice, and so on. sk Capacity of seminar kK. This is the maximum number of students that can be assigned to the seminar. We need to decide which students should be assigned to which seminars. Thus, we have the following decision variable DECISION VARIABLES Xjk1 if student jJ is assigned to seminar kK;0 if not a) Using this notation, formulate the objective of minimizing the sum or the assigned rankings. Note that this will also minimize the average assigned ranking b) Using this notation, write down the constraints that say that the maximum number of students that can be assigned to a seminar is sk c) Write down a constraint that says that each student is assigned to at least two seminars. d) Write down a constraint that says that for any given seminar a student can be assigned to the seminar at most once. e) Write down the non-negativity constraints. Problem 3: Now let us formulate the dual of the seminar assignment problem. a) Convert the formulation of problem 1 to standard form (a maximization problem with less than or equal to constraints and non-negative variables) b) With this notation, formulate the dual of the seminar assignment problem. Let k be the dual variable associated with the kth seminar capacity constraint of the problem in standard form. Similarly, let j be the dual variable associated with the jth primal student assignment constraint of the problem in standard form, meaning the constraint that says that each student must be assigned to at least two seminars. Finally, let jk be the dual variable associated with the (j,k) constraint stating that any student can be assigned to a seminar at most one time. a. What is the dual objective function? b. What do the dual constraints look like? Write them out using this notation. c) If you did part (a) of problem 2 correctly, only 7 students were assigned to the Math seminar. Given that this is the case, what is the value of the dual variable associated with this seminar? Briefly justify your answer. Note that even if you did not get parts (a) and (b) of this problem done correctly, you should be able to do this part of the problem. d) Based on the Shadow Prices in the Sensitivity Report for Problem 2, part (a), which seminar(s) should you increase to a capacity of 10 if you want to have the most impact on the average assigned ranking? You should verify this answer by resolving the problem for any case you think may have a big impact