Problem:
The toy company, Lege, produces two types of wooden toys: wooden cars and wooden planes. There are two processing lines available. In line 1, one car and two planes can be produced by using one hour of labor and 4 units of wood; in line 2, three cars and one plane can be produced by using one hour of labor and 5 units of wood. The company has 300 regular labor hours and can buy at most 50 extra hours of labor at $10/hour each month. Lege can buy wood from different companies at different prices. Overall, in each month, Lege pays $2.5/unit for the first 800 unit’s wood, $3/unit for the next 200 unit’s wood, and $4/unit for more wood. (For example, if Lege buys 1200 units wood, the total cost is 2.5 X 800 + 3 X 200 + 4 X (1200 - 800 - 200). ) The market price of a wooden car is $25 and the price of a wooden plane is $30. The company can sell any amount of toys that it produces.
Write a linear programming model to maximize Lege's monthly profit. (Define the decision variables clearly and explain the objective and constraints.)