1. (Nonzero Minimums): Referring back to the last problem, which was about mixing juices to create two fruit beverages cranapple and appleberry suppose that the profit for cranapple changes from 3 cents per gallon to 2 cents, and that the profit for appleberry changes from 4 cents per gallon to 5 cents. Suppose also that the manufacturer wants to make sure that each type of juice mixture is always produced. So, they incorporate nonzero minimums into the linear program specifications. Say they always want at least 20 gallons of cranapple produced and at least 10 gallons of appleberry. How many gallons of each type of beverage should be produced in this case in order to obtain the highest profit without exceeding available supplies?

• Regardless of which type of problem it is, your best bet in tackling a linear programming word problem is to follow this sequence of steps:

1. Read the problem carefully and identify the resources and the products.

2. Make a mixture chart, showing the resources, the products, and the profit or cost.

3. Assign an unknown quantity x or y, to each product. Use the mixture chart to write down the resource constraints, the minimum (or maximum) constraints, and the profit/cost formula.

4. Graph the feasible region.

   • The feasible region is the collection of all physically possible solution choices that can be made.

   • It always has a convex shape.

     • A shape is convex if every line segment joining any two points in the boundary is contained completely inside the shape

5. Find the coordinates of all the corner points of the feasible region.

6. Either graph the objective function (Method 1) or evaluate the profit/cost formula at each of the corner points (Method 2) to find the maximum or minimum value of the objective function.