I have the answer to A but need help with me. The additional hint that I got was the trick with 3b is that the discount can never be more than ~10%.The problem states that you can get 30% off the smallest dollar value tile order and there are three categories.Therefore, the biggest the smallest order can be is 33.3%.Multiple 33.3% x 0.3 and you get about 10%.

Hartley Homes in North Texas is building a development in Northern Tarrant County and is currently negotiating with two flooring dealers on tile prices. Both dealers offer several competing tile families consisting of small (mosaic or 1 x 1), medium (4 x 4 or subway), and large (12 x 12 and 12 x 24) tiles. Each dealer represents a different tile manufacturer, but there is reasonable overlap in styles while still permitting nuanced differences for the discriminating buyer. Both tile manufacturers deal in the high-quality ("pricey") tile market.

The dealers have offered Hartley Homes competing prices for each category of tile, but the prices are only good for 90 days. To lock in these prices for the next 32 months, Hartley must sign a contract with each dealer committing to a certain minimum dollar amount for each tile category (small, medium, large). Hartley wants to determine how many dollars to contractually guarantee each dealer to lock in these prices.

Over the next 32 months, Hartley knows that they will need, at a minimum, 40,000 total SQFT of small tile, 60,000 total SQFT of medium tile, and 240,000 total SQFT of large tile. Additionally, McBryde knows that if they order large tile from a particular dealer, they will need to purchase at least 10% of this same square footage from the dealer's medium tile category to match designs. The prices offered per square foot are given in the table below.

Prices ($/SQFT) Small Medium Large

Dealer 1 13 8 5.5

Dealer 2 12 10 6

A. Determine a minimum cost purchasing plan for Hartley. From that purchasing plan, determine a dollar amount for each category that Hartley can guarantee to each dealer.

B. Suppose Dealer 2 offers a special incentive (a "sweetener") to encourage purchases from all three of their tile categories. Dealer 2 will discount the smallest of Hartley's three purchases by category (dollars for small, dollars for medium, and dollars for large) by 30%. Incorporate this into your model, determine a new optimal purchase plan, and determine a dollar amount for each category that Hartley can guarantee to each dealer.

Hint: You only need one new decision variable but three new constraints.