What is it? Linear Programming is a method of finding the maximum or minimum value of a function that satisfies a given set of conditions called constraints. Objective Function Constraints Feasible Region
Example Problem You need to buy some filing cabinets. You know that 3-drawer filing cabinets cost $10 per unit, requires six square feet of floor space, and holds eight cubic feet of files. Also, you know 4-drawer filing cabinets cost $20 per unit, requires eight square feet of floor space, and holds twelve cubic feet of files. You have been given $140 for this purchase, though you don't have to spend that much. The office has room for no more than 72 square feet of cabinets. How many of which model should you buy, in order to maximize storage volume?
Present the Problem We are trying to find the number of each type of cabinet to buy that would give us the maximum storage volume, yet stay within our price range and the amount of room that we have in the office.
Define Your Variables Answers what you are looking for X- will be the number of 3-drawer filing cabinets that you will purchase Y- will be the number of 4-drawer filing cabinets that you will purchase
Write Objective Function What are you trying to maximize or minimize? We are trying to maximize the storage volume 3-drawer filing cabinets hold 8 Cubic Feet 4-drawer filing cabinets holds 12 Cubic Feet
State the Constraints Only $140 to make the purchase Only 72 feet of Floor Space There is nothing to give back. (so cant purchase negative numbers of either type)
Create Inequalities for the Constraints Only $140 to make the purchase 3-drawer filing cabinets cost $10 per unit 4-drawer filing cabinet costs $20 per unit
Create Inequalities for the Constraints (Cont.) Only 72 feet of Floor Space 3-drawer filing cabinets take up 6 square feet on the floor 4-drawer filing cabinets take up 8 square feet on the floor
Create Inequalities for the Constraints (Cont.) There is nothing to give back. (so cant purchase negative numbers of either type)
List of all Constraints/ Convert to Slope-Intercept form Or y < –( 1 / 2 )x + 7 Or y < –( 3 / 4 )x + 9
Graph All Constraints Feasible region (grey) is where they all overlap
Find all Vertices Around Feasible Region (0,7) (12,0) (0,0) (8,3)
Test all Vertices in the Objective Function VertexFunction (8x +12y = V) Volume (0,7) 8(0) +12(7) 84 (0,0) 8(0) + 12 (0) 0 (12, 0) 8(12) + 12(0) 96 (8,3) 8(8)+12(3) 100