Linear Programming Linear Programming ProjectProject

Warm-UpWarm-UpWrite an Inequality for the

following

What is it?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 ProblemExample ProblemYou 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 ProblemPresent the ProblemWe 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 VariablesDefine Your VariablesAnswers 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 FunctionWrite Objective FunctionWhat 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 ConstraintsState the ConstraintsOnly $140 to make the purchaseOnly 72 feet of Floor SpaceThere is nothing to give back. (so

can’t purchase negative numbers of either type)

Create Inequalities for the Create Inequalities for the ConstraintsConstraintsOnly $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 Create Inequalities for the Constraints (Cont.)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 Create Inequalities for the Constraints (Cont.)Constraints (Cont.)There is nothing to give back. (so

can’t purchase negative numbers of either type)

List of all Constraints/ List of all Constraints/ Convert to Slope-Intercept Convert to Slope-Intercept formform

Or y < –( 1/2 )x + 7

Or y < –( 3/4 )x + 9

Graph All ConstraintsGraph All ConstraintsFeasible region (grey) is where

they all overlap

Find all Vertices Around Find all Vertices Around Feasible RegionFeasible Region

(0,7)

(12,0)(0,0)

(8,3)

Test all Vertices in the Test all Vertices in the Objective FunctionObjective FunctionVertex Function

(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

Find Maximum/MinimumFind Maximum/MinimumVertex Function

(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

Maximum

State a Conclusion to your State a Conclusion to your ProblemProblem(8,3)

In order to maximize the storage space we have, it would be most beneficial for you to buy 8 3-drawer filing cabinets and 3 4-drawer filing cabinets.