Section 3-4 Objective: To solve certain applied problems using linear programming. Linear Programming

Introduction Often decisions are made based on data and are made based on certain constraints. –For example, when you decide on what cell phone wish to purchase you only have a finite amount of money, you must consider the cost of the phone and the cost of the monthly plan. –Your budget is a constraint on your decision making. –You have many cell phones to choose from, but you must choose a phone that is within your budget (constraint)

Introducti on If you ran your own pizza business you must use your resources as efficiently as possible. – Resources: Pizza Oven, Employees, Ingredients Ovens can only run for a finite amount of time Finite amount of employees Finite amount of ingredients Goal: maximize profit while working within your limits (constraints) Each of the above are constraints: a set of linear inequalities

Linear Programming – Example 1 Suppose a TV manufacturing company produces LCD and Plasma TV’s using three different machines, A, B, and C. The table below shows how many hours are required on each machine per day to produce a Plasma TV or a LCD TV. MachineLCD (x) Plasma (y) Hrs Available ABCABC 1 h 4 h 2 h 1 h 4 h x + 2y ≤ 16 Constraints: x + y ≤ 9 4x + 4y ≤ 24 x ≥ 0 y ≥ 0

Example 1 - Continued Goal: Maximize Profit – To maximize profit we have to work within our constraints (all five inequalities must be satisfied) –Recall: x + 2y ≤ 16 Constraints: x + y ≤ 9 4x + y ≤ 24 x ≥ 0 y ≥ 0

Example 1 - Continued x + 2y ≤ 16 Constraints: x + y ≤ 9 4x + y ≤ 24 x ≥ 0 y ≥ 0 (0,8) (16,0) (0,9) (9,0) (0,24) (6,0) 4x + y ≤ 24 x + 2y ≤ 16 x + y ≤ 9 Any point in this region (the common area) is called a feasible solution The set of these points is called the feasible region

Example 1 - Continued Maximizing Profit: Suppose the company makes $60 profit on each LCD TV and $40 profit on each Plasma TV. Question: how many LCD TV’s and Plasma TV’s should be produced each day to maximize profit? Profit = 60x + 40y How do we approach finding the maximum profit?

Example 1 - Continued (6,0) (5,4) (2,7) (0,8) 4x + y ≤ 24 x + y ≤ 9 x + 2y ≤ 16 Feasible Region Recall: Profit = 60x + 40y To find the maximum profit we could evaluate every point in the feasible region (all possible combinations of Plasma and LCD TVs) However, this would be VERY tedious Better Approach: 1.Plot the profit line 2.Find the points that give the greatest profit 60x + 40y Furthest point out in the feasible region (maximum) Profit = 60x + 40y Profit = 60(5) + 40(4) Profit = Profit = 460 (0,0) Note: the blue line is called the Profit Line.

Example 2 – Minimizing Cost See the example in the text on page 111

The Corner-Point Principle A maximum or minimum value of a linear expression P = Ax + By, if it exists, will occur at a corner point of the feasible region. – Note - the previous example: The maximum occurred at a corner point (5,4) The minimum occurred at a corner point (0,0)

History Linear programming arose as a mathematical model developed during WWII to plan expenditures and returns in order to reduce costs to the army and increases losses to the enemy. It was developed by George Dantzig (mathematician) It was kept secret until Postwar, many industries found its use in their daily planning.

Put very informally, LP is about trying to get the best outcome (e.g. maximum profit, least effort, etc) given some list of constraints (e.g. only working 30 hours a week, not doing anything illegal, etc), using a linear mathematical model. Overview:

Where is Linear Programming Used? Most extensively it is used in business and economic situations, but can also for engineering problems. Some industries that use linear programming models: –Transportation –Energy –Telecommunications –Manufacturing –It has proved useful in modeling diverse types of problems in planning, routing, scheduling, assignment, and design. Ex) Linear programming can be used to determine the best assignment of 70 people to 70 jobs

Homework P : 1-9 odd