Alveva Subscription Box Service

Alveva is an online beauty subscription service that sends subscribers a monthly box of four to seven curated make-up and beauty products. Alveva's monthly subscription service allows consumers to test in-trend products before purchasing them at a full price from Alveva's website. Table 1 below displays 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 Table 2. 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 2 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.

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? Write the algebraic formulation for this optimization problem and solve it using Excel or GAMS. How many unique boxes should Alveva create?

2. The CEO of Alveva would like to keep the number of different boxes created to a maximum of 5 to process boxes faster. What should be included in these 5 boxes, how should they be assigned to different subscribers, and what is the optimal profit? Write the algebraic formulation and solve using Excel or GAMS.  

3. You will find that the total profit in part 2 is less than that of part 1. To make up for this lost profit, Alveva would like to introduce a limited-edition box to be sold at 50% of its value that both subscriber and non-subscriber customers could purchase. The firm will produce 50,000 limited-edition boxes and expects to sell all. The box should not contain any products included in this month's subscriber boxes. Which products should this box include to maximize profits? Can the firm recover the lost profits? Should moisturizer D be included in this box? Explain your reasoning. The value of each product is given in Table 3.