Blending problems involve mixing several raw materials that have different characteristics to make a

product that meets certain specifications. Dietary planning, gasoline and oil refining, coal and fertilizer

production, and the production of many other types of bulk commodities involve blending. We typically

see proportional constraints in blending models.

Model and solve the following example:

F$-$Oil sells gasoline and heating oil. These products are produced by blending two types of crude oils

together. F$-$Oil has $5,000$ barrels of crude oil $1$ and $10,000$ barrels of crude oil $2$ available.

The two types of crude oil differ in quality with respect to various characteristics such as octane rating,

viscosity, and others. Suppose these different qualities of interest are collectively represented as what we

will refer to as the quality level.

Each barrel of crude oil $1$ has a quality level of $10$ and each barrel of crude oil $2$ has a quality level of $5.$

Gasoline must have an average quality level of at least $8,$ whereas heating oil must have an average

quality level of at least $6.$ Gasoline sells for $$75$ per barrel, and heating oil sells for $$60$ per barrel. In

addition, if any barrels of the crude oils are leftover, they can be sold for $$65$ and $$50$ per barrel,

respectively. We assume that demand for heating oil and gasoline is unlimited, so that all production can

be sold. F$-$Oil wants to maximize its revenue from selling gasoline, heating oil, and any leftover crude

oils.

a$)$ Develop a spreadsheet model and solve.

b$)$ Express the model in algebraic form