You have just become the manager of a plant producing plastic products. Although the plant operation involves many products and supplies, you are interested in only three of the products:

(1). A vinyl–asbestos floor covering, the output of which is measured in boxed lots, each covering a certain area; 

(2). A pure vinyl countertop, measured in linear yards; and

(3). A vinyl–asbestos wall tile, measured in squares, each covering 100 ft². Of the many resources needed to produce these plastic products, you have identified four: vinyl, asbestos, labor, and time on a trimming machine. A recent inventory shows that on any given day you have 1500 lb of vinyl and 200 lb of asbestos available for use.
Additionally, after talking to your shop foreman and to various labor leaders, you realize that you have 3 person-days of labor available for use per day and that your trimming machine is available for 1 machine-day on any given day. The following table indicates the amount of each of the four resources required to produce a unit of the three desired products, where the units are 1 box of floor cover, 1 yard of counter top, and 1 square of wall tiles. Available resources are tabulated below.

Formulate a mathematical model to help determine how to allocate resources to maximize profits.

a. Identify the decision variables: What decision is to be made?

b. Formulate the objective function: How do these decisions affect the objective?

c. Formulate the constraint set: What constraints must be satisfied? Be sure to consider whether negative values of the decision variables are allowed by the problem, and ensure they are so constrained if required.