Linear Programming: Maximizing Aid to Hurricane Victims

1. 5-Minute Check on Activity 3-5 y Solve the following inequalities: y < 3x – 2y > x + 2 Where do we shade and is the line solid or dashed? y < 2x – 1 y ≥ 3x + 2 y ≤ 2x – 6 y > x + 2 x = 2, y = 4 is corner point shade below first line and above second line x < -- shade below line and line is dashed ≥ -- shade above line and line is solid ≤ -- shade below line and line is solid > -- shade above line and line is dashed Click the mouse button or press the Space Bar to display the answers.

2. Activity 3 - 6 Helping Hurricane Victims Vanceboro, NC 10/2/2010, photo courtesy of Weather Underground

3. Objectives • Determine the objective function in a situation where a quantity is to be maximized or minimized • Determine the constraints that place limitations on the quantities contained in the objective function • Translate the constraints into a system of linear inequalities • Use linear programming method to solve a problem in which a quantity is to be maximized or minimized subject to a set of constraints

4. Vocabulary • Objective function – an equation that describes a quantity to be maximized or minimized in terms of two or more variables. • Constraints – the limitations or restrictions placed on the variables upon which the quantity to be maximized or minimized depends. Each constraint is written as a linear inequality. • Feasible region – the graph of the solution set of the system of linear inequalities representing the constraints. • Feasible points –the corner points determined by the intersection of the boundary lines of the feasible region in a linear programming problem.

5. Linear Programming (LP) Linear programming is a mathematical model used to determine the “best” way to attain a certain objective subject to a set of constraints • Objectives are usually something like maximization of output or minimization of cost • The optimal solution (“best”) to a problem is found at a corner point of the feasible region bounded by certain constraints • Constraints are the inequalities

6. y LP – A Graphical View Optimal solutions are only found at the corner points. Advanced techniques, like the simplex method, provide a way to evaluate the corner points. A graphical view is that the objective function is tangent to the feasible region at the optimal solution (see above) x • Constraints (inequalities) are laid out like last lesson • Feasible Region is drawn • Corner points are determined • Objective function is drawn and translated out until it is tangent (at a corner point)

7. Activity Food and clothing are being sent by commercial airplanes to hurricane victims in Florida. Each container of food is estimated to feed 12 people. Each container of clothing is intended to help 5 people. Organizers of the relief effort want to determine the number of containers of food and clothing that should be sent in each plane shipment that will maximize the number of victims helped. Maximize N = 12F + 5C (Objective Function)

8. Activity continued Weight and space restrictions imposed by the airlines are summarized as follows: • Total weight cannot exceed 19,000 pounds • Total volume must be no more than 8,000 cubic feet A container of food weighs 50 pounds and occupy 20 cubic feet. A container of clothes weighs 20 pounds and occupy 10 cubic feet. Write the constraints as a linear inequality: 50F + 20C ≤ 19000 Weight constraint 20F + 10C ≤ 8000 Volume constraint

9. Activity continued Logical restrictions as follows: • You can’t ship negative containers of food • You can’t ship negative containers of clothes Logical constraints help to limit the feasible region; after all we are trying to help the help and bringing nothing with you to help makes no sense. Write these constraints as a linear inequality: F > 0 Food constraint C > 0 Clothes constraint

10. Activity continued Writing all that we have figured out in a mathematical problem format: • Maximize N = 12F + 5C Objective Function subject to the following constraints: • 50F + 20C ≤ 19000 Weight constraint • 20F + 10C ≤ 8000 Volume constraint • F ≥ 0 Food constraint • C ≥ 0 Clothes constraint

11. Graphical Method to Solve • Graph the Constraints (inequalities) • Determine the feasible region • Locate corner points (need the values!)Usually the intercepts and intersection • Objective function is drawn and translated out until it is tangent (at a corner point) ORUse a table to evaluate all the corner points

12. Solving the Hurricane Relief Problem • 50F + 20C ≤ 19000 Weight constraint • 20F + 10C ≤ 8000 Volume constraint • F ≥ 0 Food constraint • C ≥ 0 Clothes constraint 50F + 20C ≤ 19000 20F + 10C ≤ 8000 20C ≤ 19000 – 50F 10C ≤ 8000 – 20F C ≤ 950 – (5/2)F C ≤ 800 – 2F Y1 = 950 – (5/2)X (don’t graph F>0 and C>0) Y2 = 800 – 2X Window(-1, 500, 50 -1, 1000, 50)

13. C F Solving the Hurricane Relief Problem C1 = 950 – (5/2)F F>0 and C>0 C2 = 800 – 2F Determine corner points: C1: y-intercept (0, ) x-intercept ( , 0) C2: y-intercept (0, ) x-intercept ( , 0) Intersection point: ( , )

14. C F Solving the Hurricane Relief Problem C1 = 950 – (5/2)F F>0 and C>0 C2 = 800 – 2F Determine corner points: C1: y-intercept (0, 950) x-intercept (380, 0) C2: y-intercept (0, 800) x-intercept (400, 0) Intersection point: (300, 200)

15. Finding Corner Points • Y-intercepts: read off of y = mx + b • X-intercepts: set y = 0 and solve 0 = -2.5x + 950 0 = -2x + 800 2.5x = 950 2x = 800 x = 380 x = 400 • Intersection Point: set equations equal -2.5x + 950 = -2x + 800 950 = 0.5x + 800 150 = 0.5x 300 = x y = -2(300) + 800 = 200

16. Solving the Hurricane Relief Problem Maximize N = 12F + 5C C1 = 950 – (5/2)F F>0 and C>0 C2 = 800 – 2F Evaluating Objective Function at corner points: Corner point (300, 200) maximizes the objective function!

17. Summary and Homework • Summary • Linear programming is a method used to determine the maximum or minimum values of a quantity that are dependent upon variable quantities that are restricted. • A linear programming problem consists of an objective function and a set of constraints. • Homework • pg 358 – 364; 1 and 2