Section 7.6 Linear Programming
Objective Functions in Linear Programming
We will look at the important application of systems of linear inequalities. Such systems arise in linear programming, a method for solving problems in which a particular quantity that must be maximized or minimized is limited by other factors. Linear programming is one of the most widely used tools in management science. It helps businesses allocate resources to manufacture products in a way that will maximize profits. An objective function is an algebraic expression in two or more variables describing a quantity that must be maximized or minimized.
Example An adjunct college professor makes $12 an hour tutoring at a local tutoring center, and $24 an hour teaching at the local communicty college. Write an objective function that describes total weekly earnings if hours worked tutoring is x and hours worked teaching is y.
Example A company manufactures kayaks and canoes. If the company's profits are $200 on the kayaks and $150 on canoes, write the objective equation for the profit, z, made on x kayaks and y canoes that can be produced in one month.
Constraints in Linear Programming
Example The truck that takes the canoes and kayaks from the factory can carry only a limited number with cargo space of 750 cubic feet. If each kayak takes up 5 cubic feet and each canoe takes up 15 cubic feet of space, write an equation that describes this situation where x is the number of kayaks and y is the number of canoes. This is a constraint for the manufacturing company.
Example The truck that carries the kayaks and canoes can only carry a maximum of 4000 lbs. The kayaks and canoes both weigh the same amount, 40 lbs. If x is the number of kayaks and y is the number of canoes, write an equation that describes this situation. This is another constraint that the manufacturing company must consider.
Solving Problems with Linear Programming
Example Step 1: A: Write down the Objective equation that exists for the kayak/canoe manufacturing company. B: Write the two constraint inequalities that you found in the previous two examples. C: Graph the two constraints on the same graph and note the intersection of the regions. y Continued on the next screen. x
Example Step 2: A: From step 1 you should have the graph shown below. B: Now locate the vertices of this region by finding the point of intersection of the two lines, and the x and y intercepts. Put these points in the chart below and plug those points into the objective equation to find which vertex gives you the maximum profit. Corner (x,y) Objective: 200x+150y=z
Example A plane carrying relief food and water to a tidal wave ravaged community can carry a maximum of 50,000 lbs, and is limited in space to carrying no more than 6000 cubic feet. Each container of water weighs 60 lbs and takes up 1 cubic foot and each food container weighs 50 lbs and takes up 10 cubic feet of space. The relief organization wants the plane to assist as many people as possible and it is known that the water containers can take care of 4 people and the food containers can feed 10 people. Draw the region of constraint and make recommendations on how many containers of water and food should be taken on the plane. How many people will get food? water? Corner (x,y) Objective:
Example An adjunct professor makes $12/hour tutoring at a local tutoring center and $24/hr teaching at the local community college. Let x be the number of hours tutoring and y the number of hours teaching. You have already written the objective equation in a previous problem. There are however, more constraints on her time. In order to take care of her children, she can only work 20 hours a week, and the college requires that she teach at least 5 hours a week for them, but no more than 8 hours. How many hours should she work tutoring and teaching. What will be her maximum income each week? Corner (x,y) Objective:
A farmer grows peaches (p) and apples(a). He knows that to prevent a certain pest infection, the number of peach trees cannot exceed the number of apple trees. Also because of space requirements for each tree, the number of peach trees plus twice the number of apple trees cannot exceed 100 trees. He wants to produce the maximum number of bushels of fruit from his orchard and he knows that each peach tree produces 80 bushels and apple trees each produce 100 bushels. What is the constraint for space requirements. (a) (b) (c) (d) p a p 2a 100 80 p 100a z 80a 100 p z
A farmer grows apples and peaches. He knows that to prevent a certain pest infection, the number of peach trees cannot exceed the number of apple trees. Also because of space requirements for each tree, the number of peach trees plus twice the number of apple trees cannot exceed 100 trees. He wants to produce the maximum number of bushels of fruit from his orchard and he knows that each peach tree produces 80 bushels and apple trees each produce 100 bushels. What is the objective function for this example. (a) (b) (c) (d) p a p 2a 100 80 p 100a z 80a 100 p z