EXAMPLE 3 Revisiting Our Clothing Manufacturer
We can also translate the information in the mixture chart shown in Figure 4.2 into inequalities and an equation for expressing the profit in terms of how many shirts and vests are produced. Using the first column of the mixture chart and the fact that only 60 units of cloth are available, we can write
And using the last column, we get the following expression for the profit :
Now that we have the information from the original problems represented in mathematical terms, we will return our attention to finding a solution to the problems.
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Finding the best (largest profit) mixture of skateboards and/or dolls to make can be carried out in two phases.
Feasible Set or Feasible Region DEFINITION
The feasible set, also called the feasible region, for a linear-programming problem is the collection of all physically possible solution choices that can be made.
We can use a geometric diagram such as the one in Figure 4.3 to help us understand the feasible set of options that the manufacturer of skateboards and dolls has available. The geometric diagram we draw will have as many “dimensions” as there are products being manufactured. We have two products represented by the variables and , so we use a two-dimensional picture. Even diagrams involving three variables are hard to draw and visualize. Though these diagrams helped with developing algorithms for solving mixture problems, they are of little practical use for realistic problems.