# 10.6.3. Linear Programing and Optimization

A workshop of two workers produces two types of iron desks; one of the workers builds the desks and the other one sprays them. It takes the first worker 4 hours to build a unit of the first type and 3 hours to build a unit of the second type, while it takes the second worker 3 hours to spray a unit of the first type and 4 hours to spray a unit of the second type. The first person works at least 5 hours a day, and the other works a maximum of 7 hours a day. If the workshop earns a profit of LE from each single unit, determine the relations required for calculating the number of units of the first type and the number of units of the second type to be produced every day from each type to maximize the profit .

• A, , , ,
• B, , , ,
• C, , , ,
• D, , , ,

### Example

A workshop of two workers produces two types of iron desks; one of the workers builds the desks and the other one sprays them. It takes the first worker 4 hours to build a unit of the first type and 3 hours to build a unit of the second type, while it takes the second worker 3 hours to spray a unit of the first type and 4 hours to spray a unit of the second type. The first person works at least 5 hours a day, and the other works a maximum of 7 hours a day. If the workshop earns a profit of LE from each single unit, determine the relations required for calculating the number of units of the first type and the number of units of the second type to be produced every day from each type to maximize the profit .

### Solution

Suppose is the number of the first type of iron desk, and is the number of the second type. Since it is impossible to build a negative number of the first type or the second type, both and must be greater-than-or-equal-to 0.

When referring to the first worker, the words "at least" indicate that the sum of and is greater-than-or-equal-to 5, and when referring to the second worker, the word "maximum" indicates that the sum of and is less-than-or-equal-to 7.

Since the profit on the first type is LE, the total profit on all of the first type sold is LE, and since the profit on the second type is LE, the total profit on all of the second type sold is LE. Therefore, the total profit, , on all the desks sold is LE.

This means that the relations required for calculating the number of desks to be produced each day to maximize profit are , , , , .

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