Chapter 3.8, Problem 5P

Explanation of Solution

 Formulation of a Linear Programming (LP) to help Chandler maximize the profit:

• Let “x_ij” be the barrels of oil “i” used to make product “j”, which means i = 1 refers to oil 1, i = 2 refers to oil 2, j = 1 is gasoline, and j = 2 is heating oil.
• Here, oils are inputs and gasoline and heating oils are outputs.
• Let “y_j” be the number of dollars spent advertising the product “j”.
• The “x₁₁” means the number of barrels of oil 1 used to produce gasoline and “x₁₂” means the number of barrels of oil 1 used to produce heating oil.
• Likewise, “x₂₁” means the number of barrels of oil 2 used to produce gasoline and “x₂₂” means the number of barrels of oil 2 used to produce heating oil.
• Therefore, “$x_{11}+x_{12}$” refers the number of barrels of oil 1 used and “$x_{21}+x_{22}$” will be total number of barrels of oil 2 used.
• Also, “$x_{11}+x_{21}$” refers the total production of gasoline and “$x_{12}+x_{22}$” refers the total production of heating oil.

 Constraint 1:

 The $25 profit is given for each barrel of gasoline. Then the total profit on “$x_{11}+x_{21}$” barrels of gasoline is $\$25\left(x_{11}+x_{21}\right)$

 Likewise, the total profit on “$x_{12}+x_{22}$” barrels of heating oil is given as $\$20\left(x_{12}+x_{22}\right)$

 Since $\left(y_1+{\text{ y}}_2\right)$ is total advertisement cost on both products, the net profit function of the problem is shown as follows:

 $z=25\left(x_{11}+{\text{ x}}_{21}\right)+20\left(x_{12}+{\text{ x}}_{22}\right)-y_1-y_2$

 As the objective of problem is to maximize the net profit, the objective function is given as follows:

 $\text{Max Z}=25\left(x_{11}+{\text{ x}}_{21}\right)+20\left(x_{12}+{\text{ x}}_{22}\right)-y_1-y_2$

 Constraint 2:

 The total availability of oil 1 is 5000 barrels. Therefore, the total number of barrels of oil 1 used must be less than 5000.

 That is, $x_{11}+{\text{ x}}_{12}\le 5000$

 Likewise, the constraint for oil 2 is $x_{21}+{\text{ x}}_{22}\le 10000$

 Constraint 3 and 4:

 The average quality level of each product is defined as the ratio of total quality value of product and the total quality produced.

 Since the quality level of each oil is 10 and 5, the total quality level of gasoline is $10x_{11}+{\text{ 5x}}_{21}$. It is stated that the minimum average quality level of gasoline must be 8.

 That is,

 $\begin{array}{c}\frac{\text{Total quality of oil used for gasoline}}{\text{Total amount of gasoline produced}}\ge 8\\ \frac{10x_{11}+{\text{ 5x}}_{21}}{x_{11}+{\text{ x}}_{21}}\ge 8\\ 2x_{11}+{\text{ 3x}}_{21}\ge 0\end{array}$

 Likewise, the minimum average quality level of heating oil is 6