# 10.6.3. Linear Programing and Optimization

A factory produces two types of iron sheet offices. One of the workers builds the desks and another one sprays them. It takes the first worker 3.5 hours to build a unit of the first type and 2 hours to build a unit of the second type, while it takes the second worker 4 hours to spray a unit of the first type and 2 hours to spray a unit of the second type. The first worker works at least 5 hours a day and the other one works a maximum of 8 hours a day. If the factory earns a profit of LE from each single unit, determine the number of units to be produced every day from each type in order to maximize the profit.

• A first type: 0 units, second type: 4 units
• B first type: 2 units, second type: 0 units
• C first type: 0 units, second type: 2 units
• D first type: 4 units, second type: 0 units

### Example

A factory produces two types of iron sheet offices. One of the workers builds the desks and another one sprays them. It takes the first worker 3.5 hours to build a unit of the first type and 2 hours to build a unit of the second type, while it takes the second worker 4 hours to spray a unit of the first type and 2 hours to spray a unit of the second type. The first worker works at least 5 hours a day and the other one works a maximum of 8 hours a day. If the factory earns a profit of LE from each single unit, determine the number of units to be produced every day from each type in order to maximize the profit.

### Solution

Since the profit on the first type of iron sheet office is LE, the total profit on all of the first type is LE, and since the profit on the second type of iron sheet office is LE, the total profit on all of the second type is LE. Therefore, if the total profit on all the iron sheet offices is , the objective function is . The vertices of the feasible region are , , , and approximately . First, find the value of the objective function at the vertex .

Next, find the value of the objective function at the vertex .

Now find the value of the objective function at the vertex .

Finally, find the value of the objective function at the vertex of approximately .

Thus, finding the value of the objective function at each vertex shows that the function has a maximum at . This means that to maximize profit, the factory should produce 0 units of the first type of iron sheet office each day, and 4 units of the second type.

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