Alveva is an online beauty subscription service that sends its subscribers a monthly box of four to seven curated make-up and beauty products. Alveva's monthly subscription service allows customers to test in-trend products before purchasing them at a full price from Alveva's website. Table 1 (second column) shows the per unit wholesale cost of each product that can be used to create different boxes each month.

After customers receive their subscription boxes, they can indicate the products they did not enjoy through a monthly survey. The Analytics Department at Alveva uses this information to categorize customers into 10 subscriber types. The number of customers that belong to each subscriber type is given in the last row of Table 1. In addition to collecting subscribers' feedback, the firm also keeps track of the products that the customers receive each month to ensure the delivery of new items in each box. The ones in Table 1 indicate a product that a subscriber type must not receive in their next box. For example, subscriber type 1 must not receive moisturizer A, lipstick C, concealer B, highlighters A and B, and nail polish D in their next monthly box. Assume the same box should be sent to all subscribers of the same type. At most 80000 units of each product type can be used in the subscription boxes each month due to resource limits.

1. If Alveva charges $8 per item in the monthly subscription box, how should the boxes be curated and assigned to different subscribers to maximize the total profit?
   1. Write the algebraic formulation for this optimization problem.
   2. Solve using Excel or GAMS and report your findings.
2. The CEO of Alveva would like to limit the number of different boxes created to a maximum of 3 to process boxes faster. In other words, at most 3 types of boxes will be produced, and each box type will be sent to multiple subscriber types (e.g. box type 1 may be sent to subscriber types 1, 2, 5, and 8). What should be included in these 3 box types, how should they be assigned to different subscribers, and what is the optimal profit? Solve using Excel or GAMS.

Hint for part 2: A (practically) desirable model for this part is a mixed-integer linear program. For this model, you need to introduce a binary variable with three indices (for example, define binary variable