Question: Blending Problem

A homeowner wants to paint his house. It has to be a one-coat job. To satisfy this requirement, the paint must have a viscosity of at least 200 centipoises. Another requirement is that, for the desired level of brilliance, there must be at least 14 g of a chemical ingredient A in each gallon of the paint. In addition, for a desired degree of durability, at least 30 g of another chemical B must be present in each gallon of paint.

There are two kinds of paint (X₁ and X₂) available to him. Type X₁ costs $6 per gallon, and type X₂ costs $4 per gallon. Their specifications include the following:

The homeowner decides to blend the two paints. He wants to make sure that combining the two paints yields at least 1 gallon of paint.

a. Present the linear program for this problem (including all constraints).

b. How much of X₁ and X₂ should be used in each gallon of the blend? Within what range of objective coefficient values does this solution remain unchanged? This is not an integer programming problem.

c. Why does reduced cost equal zero for the two variables?

d. Which constraints are binding? Which constraints are non-binding?

e. Present the dual (shadow) prices of the constraints that are binding and interpret.