1. "A production line equipment manufacturer is planning for next year's production.  There are three types of products, A, B, and C.  They cost $250,000, $300,000, and $450,000 per unit respectively, including their share of operating and overhead costs.  Sales contracts for A, B, and C are estimated as $280,000, $336,000, and $504,000, respectively.  The company has a total budget including cash and credit of $45,000,000 for next year.  Market research shows that their products need to include a minimum of 15% type “A”, 25% type “B”, and 25% type “C”.  All manufactured units must be complete.  They do not produce a fraction of equipment using the available time and budget.  The maximum annual production capacity is 120 units of all types.

Develop a model for this problem and find the optimal solution to maximize the profits.

How many units of “A” should be built?"

A) Less than 15
B) 20 or more but less than 25
C) 30 or more
D) 15 or more but less than 20
E) 25 or more but less than 30
find the optimal solution to maximize profits, we need to formulate a linear programming model. Let:

x1: number of units of type A produced
x2: number of units of type B produced
x3: number of units of type C produced

"The objective function is to maximize the total profit:    
Maximize 280,000x1 + 336,000x2 + 504,000x3"

Subject to the following constraints:

Cost constraint: 250,000x1 + 300,000x2 + 450,000x3 ≤ 45,000,000
Production capacity constraint: x1 + x2 + x3 ≤ 120
Type A minimum constraint: x1 ≥ 0.15(x1 + x2 + x3)
Type B minimum constraint: x2 ≥ 0.25(x1 + x2 + x3)
Type C minimum constraint: x3 ≥ 0.25(x1 + x2 + x3)

We can simplify the minimum percentage constraints by rearranging them:

0.85x1 - 0.15x2 - 0.15x3 >= 0
0.75x2 - 0.25x1  - 0.25x3 >= 0
 0.75x3  - 0.25x1 - 0.25x2 >= 0
x1 + x2 + x3

2. What is the optimal profit of manufacturing?

A) 5,100,000 or more but less than 5,200,000
B) 5,000,000 or more but less than 5,100,000
C) Less than 5,000,000
D) 5,200,000 or more but less than 5,300,000
E) 5,400,000 or more
F) 5,300,000 or more but less than 5,400,000