Two oil refineries each produce three grades of petroleum, AA, BBand CC.

At each refinery, the various grades of petroleum are produced in a single operation so that they are in fixed proportions.
One operation at Refinery 1 produces 2 units of AA, 1 unit of BBand 5 units of CC.
One operation at Refinery 2 produces 5 units of AA, 1 unit of BBand 2 units of CC.
Refinery 1 charges $350 for one operation, and Refinery 2 charges $200 for one operation.
A consumer needs at least 540 units of AA, 180 units of BBand 420 units of CC.
How should the orders be placed if the consumer’s needs are to be met most economically?

(a) What can the manager ask for directly?

Amount of BB

Amount of AA

Number of operations at Refinery 1

Amount of CC

Number of operations at Refinery 2


The manager orders xxoperations at Refinery 1 and yyoperations at Refinery 2.
(b) Enter the total amount of AAproduced, as a function of xxand yy.
 

(c) Enter the total amount of BBproduced, as a function of xxand yy.
 

(d) Enter the total amount of CCproduced, as a function of xxand yy.
 

(e) Each grade of oil, AA, BBand CC, gives an inequality in xxand yy. Plot these inequalities on a graph. Enter the coordinates of the corners of the feasible region (the feasible basic solutions). Enter them in increasing order of their xx-coordinate. For example, if one feasible basic solution is x=1,y=2x=1,y=2; another is x=5,y=0x=5,y=0and a third is x=2,y=3x=2,y=3, you would enter (1,2), (2,3), (5,0)

### Part (a) What can the manager ask for directly?

The manager can directly ask for:

- The **number of operations at Refinery 1** (denoted by $ x $).
- The **number of operations at Refinery 2** (denoted by $ y $).

These numbers will then determine how much of each grade of petroleum (AA, BB, and CC) is produced.

### Part (b) Total amount of AA produced

Each operation at Refinery 1 produces 2 units of AA, and each operation at Refinery 2 produces 5 units of AA. So, the total amount of AA produced is given by:

\[
AA(x, y) = 2x + 5y
\]

### Part (c) Total amount of BB produced

Each operation at Refinery 1 produces 1 unit of BB, and each operation at Refinery 2 produces 1 unit of BB. Therefore, the total amount of BB produced is:

\[
BB(x, y) = x + y
\]

### Part (d) Total amount of CC produced

Each operation at Refinery 1 produces 5 units of CC, and each operation at Refinery 2 produces 2 units of CC. Hence, the total amount of CC produced is:

\[
CC(x, y) = 5x + 2y
\]

### Part (e) Inequalities for each grade of petroleum

The consumer needs at least:

- 540 units of AA,
- 180 units of BB,
- 420 units of CC.

This gives us the following system of inequalities based on the production from $ x $ operations at Refinery 1 and $ y $ operations at Refinery 2:

1. For AA: 
   \[
   2x + 5y \geq 540
   \]
2. For BB: 
   \[
   x + y \geq 180
   \]
3. For CC: 
   \[
   5x + 2y \geq 420
   \]

### Finding the feasible region

To solve this graphically, we treat the inequalities as equations and plot them on the $ xy $-plane:

1. For $ 2x + 5y = 540 $, the intercepts are $ (270, 0) $ and $ (0, 108) $.
2. For $ x + y = 180 $, the intercepts are $ (180, 0) $ and $ (0, 180) $.
3. For $ 5x + 2y = 420 $, the intercepts are $ (84, 0) $ and $ (0, 210) $.

Next, we solve the system to find the corners (vertices) of the feasible region by finding the intersection points of these lines.

1. Intersection of $ 2x + 5y = 540 $ and $ x + y = 180 $:
   Solving these equations, we get $ x = 150 $ and $ y = 30 $.
   \[
   (150, 30)
   \]

2. Intersection of $ 2x + 5y = 540 $ and $ 5x + 2y = 420 $:
   Solving these, we get $ x = 180 $ and $ y = 36 $.
   \[
   (180, 36)
   \]

3. Intersection of $ x + y = 180 $ and $ 5x + 2y = 420 $:
   Solving these, we get $ x = 120 $ and $ y = 60 $.
   \[
   (120, 60)
   \]

4. Finally, the intercept $ (270, 0) $ on the $ 2x + 5y = 540 $ line also serves as a boundary point.

### Feasible basic solutions (in increasing order of $ x $-coordinate):

The feasible basic solutions are:

\[
(120, 60), (150, 30), (180, 36), (270, 0)
\]

### Next Steps

The next step would be to minimize the cost function, which is $ 350x + 200y $, within the feasible region. You can evaluate the cost at each of these vertices to find the most economical solution.

Would you like me to help find the optimal solution?

---

Here are five related questions:
1. What is the cost function for the consumer's problem?
2. How do we interpret the inequality constraints in a real-world scenario?
3. How can we use the vertices of the feasible region to minimize cost?
4. What happens if the consumer needs a different proportion of AA, BB, and CC?
5. How can linear programming methods like the simplex algorithm help in this scenario?

**Tip**: Always verify the corner points of the feasible region before calculating the optimal solution.

Two oil refineries each produce three grades of petroleum, AA, BB, and CC. At each refinery, the various grades of petroleum are produced in a single operation so that they are in fixed proportions. One operation at Refinery 1 produces 2 units of AA, 1 unit of BB, and 5 units of CC. One operation at Refinery 2 produces 5 units of AA, 1 unit of BB, and 2 units of CC. Refinery 1 charges $350 for one operation, and Refinery 2 charges $200 for one operation. A consumer needs at least 540 units of AA, 180 units of BB, and 420 units of CC. How should the orders be placed if the consumer’s needs are to be met most economically?

This problem involves optimizing orders between two refineries to meet specific production needs for three grades of petroleum—AA, BB, and CC—using linear programming and graphical methods. Learn how to apply system of inequalities and minimize costs based on the refinery's fixed proportions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation