Question:

Vicky is attending a summer school where she must study four units. She must do a one-hour lecture for each of the units every day. There are 6 one hour time slots in a day. As there are way too many students, repeating lectures for each unit is offered at every time slot of the day and are taught by different lecturers. Vicky’s decision on the class of a unit she likes to take is purely influenced by how much the lecturer looks like Keanu Reeves.

[Hint: the decision variables should help you decide which class Vicky will take for each unit—i.e., Vicky to take Class i of Unit k. Furthermore, the score for how much the lecturer for Class i of Unit k looks like Keanu Reeves can be represented by the score $\displaystyle p_{ik}$.]

a. Now, formulate an integer programming model to help Vicky choose

her classes that maximize the total Keanu Reeves lookalike score.

b. Derive a constraint so that Vicky never has to do more than two

consecutive classes without a break.

c. Modify the objective function, if Vicky’s objective is now to start her

day as late as possible.

My Answer:

Part A.Let $\displaystyle x_{ik}$ = 1 if she take class i of unit k, otherwise 0.

max z = $\displaystyle \sum x_{ik}p_{ik} , k=1,2,3,4. , i = 1,2,3,4,5,6.$

s.t $\displaystyle \sum x_{ik} = 1 , k=1,2,3,4 $

$\displaystyle x_{ik} \in {0,1}, i = 1,2,3,4,5,6 , k=1,2,3,4$

Part B. $\displaystyle x_{i} + x_{i+1} \leq M(1-y)$

and $\displaystyle x_{i+2} \leq My$

where $\displaystyle y \in {0,1} , M is a large integer$

Part C.Unfortunately i can't think of how to change the objective function for part c.

I'm assuming it has something to do with the 'i' values as these are relating to the time of the class but i can't figure out how to translate that into an answer.

Any help would be greatly appreciated.