Industries that use linear programming models include transportation, energy, telecommunications, and manufacturing.

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1 Linear programming arose as a mathematical model developed during the second world war to plan expenditures and returns in order to reduce costs to the army and increase losses to the enemy. It was kept secret until Postwar, many industries found its use in their daily planning. Linear programming can be applied to various fields of study. It is used in business and economics, but can also be utilized for some engineering problems. Industries that use linear programming models include transportation, energy, telecommunications, and manufacturing. It has proved useful in modeling diverse types of problems in planning, routing, scheduling, and design. 1

2 Linear Programming: Maximizing or minimizing a linear function subject to linear constraints. The constraints may be equalities or inequalities. It will determine the way to achieve the best outcome (such as maximize a profit or lower a cost) in a given mathematical model. Objective Function: Algebraic expression in 2 or more variables describing a quantity that has to be maximized/minimized. Constraints: limitations or restrictions The Solution occurs at one of the vertex points or corner points, along the boundary of the region. Find the maximum & minimum values of the objective function f = 5x+8y, subject to the constraints given by the system of inequalities. 2x+y 10 2x+3y 14 x 0 y 0 Find Corners of feasible region. Substitute ordered pairs into objective function. 2

3 Find the values that maximize the following objective function: Max = x + y Constraints: 4x + 2y 12, x + 2y 4, -x + y 1 Solving a Linear Programming Problem: 1) Define variables and write an objective function RECALL Obj. Funct: An algebraic expression in 2 or more variables describing quantity that must be maximized or minimized. 2) Write inequalities that represent the constraints (limitations). Graph & shade the solution set. 3) State all order pairs of the vertices of the feasible region (shaded region) and test into the objective function. 4) Write a statement that summarizes your answer. 3

4 On June 24, 1948, the former Soviet Union blocked all land and water routes through East Germany to Berlin. A gigantic airlift was organized using American and British planes to supply food, clothing and other supplies to more than 2 million people in West Berlin. The cargo capacity was cubic feet for an American plane and cubic feet for a British plane. To break the Soviet blockade, the Western Allies had to maximize cargo capacity, but were subject to the following restrictions: No more than 44 planes could be used. The larger American planes required 16 personnel per flight, double that of the requirement for the British planes. The total number of personnel available could not exceed 512. The cost of an American flight was $9000 and the cost of a British flight was $5000. Total weekly costs could not exceed $300,000. Find the number of American and British planes that were used to maximize cargo capacity. Homework #1 from worksheet 4

5 1. Bottled water and medical supplies are to be shipped to victims of an earthquake by plane. Each container of bottled water will serve 10 people and each medical kit will aid 6 people. The planes that ship these supplies are subject to weight and volume restrictions. Each plane can carry no more than 80,000 pounds. The bottled water weighs 20 pounds per container and each medical kit weighs 10 pounds. In addition to a weight constraint on its cargo, each plane has a limited amount of space in which to carry supplies. Each plane can carry a total volume of supplies that does not exceed 6000 cubic feet. Each water bottle and medical kit has a volume of 1 cubic foot. Determine how many bottles of water and how many medical kits should be sent on each plane to maximize the number of earthquake victims who can be helped. 1. A student earns $10 per hour for tutoring and $7 per hour as a teacher s aide each week. a) Write an objective function that describes total weekly earnings. b) The student is bound by the constraints listed below. Write an inequality for each of the following: To have enough time free time, the student can work no more than 20 hours per week. The tutoring center requires that each tutor spend at least 3 hours per week tutoring. The tutoring center requires that each tutor spend no more than 8 hours per week tutoring. c) Use linear programming to find the number of hours this student should spend in order to maximize her profit. 5

6 2. A manufacturer produces two models of mountain bikes. The times (in hours) required for assembling and painting are given in the following table: Model A Assembling 5 4 Painting 2 3 Model B The maximum total weekly hours available in the assembly department and the paint department are 200 hours and 108 hours respectively. The profits per unit are $25 for model A and $15 for model B. How many of each type should be used to maximize profit? 6

7 You are about to take a test that contains computation problems worth 6 points each and word problems worth 10 points each. You can do a computation problem in 2 minutes and a word problem in 4 minutes. You have 40 minutes to take the test and answer no more than 12 problems. Assuming you answer all of the problems answered correctly, how many of each type of problem must you answer to maximize your score? What is the maximum score? 7