4. (15 pts) Modeling with binary variables. A business sells video games and movies. For each video game sold, the business earns a profit of $1. For each movie sold, the business earns a profit of $2. The store manager, who is a bit quirky, has imposed the following restrictions on customers' purchases. i) There is a limit of eight items (total) per customer. ii) The number of movies purchased minus the number of video games purchased cannot exceed 2. iii) The number of video games purchased minus the number of movies purchased cannot exceed four. iv) A customer can purchase zero, one, four, or six video games. No other quantities of video games are allowed. There is no such restriction on the quantity of movies. The manager would like to determine the combination of purchases by a customer that will maximize profit. Letting xi and 22 represent the quantities of video games and movies, respectively, the problem can be formulated as follows. 2 11 21 + + maximize + 2.02 subject to 22 0, and integer 21 = 0,1,4, or 6 (a) Reformulate the problem as an equivalent integer linear program. Note: The formulation given above is not linear in the decision variables because of the last constraint. 2] = 0,1.4, or 6 Hint: Introduce binary decision variables, one for each possible value of 21. (b) How would your answer to part 4a change if the objective function was instead maximize z= x} + 2x2