3.4 Linear Programming Rita Korsunsky

Example: Maximizing a Profit A small TV manufacturing company produces console and portable TV’s using three different machines, A, B, and C. The table below shows how many hours are required on each machine per day in order to produce a console TV or a portable TV. Company makes a $60 profit on each console TV and a $40 on each portable TV. How many of each should be produced every day to maximize the profit? MachineConsolePortableHours available A1 h2 h16 B1 h 9 C4 h1 h24

x06 y85 x34 y65 x45 y84 MachineConsolePortableHours available A1 h2 h16 B1 h 9 C4 h1 h24

Plot several profit lines: 03y 20x x06 y90 As profit increases, the corresponding profit lines move farther away from the origin. A company makes a $60 profit on each console TV and a $40 on each portable TV. How many of each should be produced every day to maximize the profit?

Exclude any profit value > 460 because these profit lines don’t have points in the region. “greatest-profit line” w/points in region: maximum profit: $460 (5 console TV’s and 4 portable TV’s) “least-profit line”: through corner point (0,0). In general, the greatest or least value of a linear expression will always occur at a corner point of the region. So… 2 nd method of finding Max

Example: Minimizing a Cost Every day Rhonda Miller needs 4 mg of vitamin A, 11 mg of vitamin B, and 100 mg of vitamin C. Either of two brands of vitamins can be used: Brand X at 6¢ a pill or Brand Y at 8¢ a pill. A Brand X pill supplies 2 mg of vitamin A, 3 mg of vitamin B, and 25 mg of vitamin C. A Brand Y pill supplies 1 mg of vitamin A, 4 mg of vitamin B, and 50 mg of vitamin C. How many pills of each brand should she take each day in order to satisfy the minimum daily need most economically? Brand XBrand YMinimum Daily Need Vitamin A2 mg1 mg4 mg Vitamin B3 mg4 mg11 mg Vitamin C25 mg50 mg100 mg Cost per pill6¢8¢

Brand XBrand YMinimum Daily Need Vitamin A2 mg1 mg4 mg Vitamin B3 mg4 mg11 mg Vitamin C25 mg50 mg100 mg Cost per pill6¢8¢ x02 y40 x13 y20.5 x04 y20

Method 2: Evaluate the cost at each corner point: (0,4) (1,2) (3,0.5) (4,0) ¢ ¢ ¢ ¢ x04 y30 Method 1: Continue graphing parallel lines until you reach the last corner point in the region. The solutions are (1,2) and (3,0.5), but it is inconvenient to take half a pill. The best choice is 1 Brand X pill and 2 Brand Y pills. Brand X is at 6¢ a pill and Brand Y at 8¢ a pill. How many of each should she take to minimize the cost?