Warm-upWarm-up Sketch the region bounded by the system of inequalities: 1) 2) Sketch the region bounded by the system of inequalities: 1) 2)

During World Mathematical Year 2000, a sequence of posters designed at the Isaac Newton Institute for Mathematical Sciences was displayed month by month in the trains of the London Underground. The posters were designed to stimulate, fascinate - even infuriate! But most importantly that they bring maths to life, illustrating the wide applications of modern mathematics in all branches of science - physical, biological, technological and financial.World Mathematical Year 2000

Optimization – find maximum (or minimum) of a function. Non-Linear Optimization (Section 2.7). Linear Optimization (Section 10.8) 1. Optimization - Summary Definition Definition: Feasible region is the bounded region defined by the constraints.

Continued… Maximize: z = 2x + y Objectivefunction Objective function: Maximize: Constraints Constraints : Subject to: Objectivefunction Objective function: Maximize: Constraints Constraints : Subject to: 2. A Linear Programming Problem

3. Linear Programming Theorem: If a Linear Programming Problem has a unique solution, the solution is located at a corner point (vertex) of the feasible region. A non-unique solution will lie along the boundary

4. Corner Point Method Solving a Linear Programming Problem 4. Corner Point Method Solving a Linear Programming Problem 1)Graph feasible region from constraints 2)Determine Corner Points of feasible region 3)Evaluate objective function at each corner point 4)Determine max/min for the problem 5)If solution is non-unique it will include the entire boundary between 2 corner points

Continued… Maximize: z = 2x + y Objectivefunction Objective function: Maximize: Constraints Constraints : Subject to: Objectivefunction Objective function: Maximize: Constraints Constraints : Subject to: 5. A Linear Programming Problem

Continued… Maximize: z = 2x + y Corner PointValue of Objective Function

Practice… p. 821 #9 maximize : subject to: maximize : subject to:

6. Application Solve a Linear Programming Problem in 2 variables 6. Application Solve a Linear Programming Problem in 2 variables Step 1: Step 1: Define variables Step 2: Step 2: Write the objective function z=Ax+By Step 3: Step 3: List restrictions (constraints) as inequalities Step 4: Solve using Linear Programming Methodolgy Step 1: Step 1: Define variables Step 2: Step 2: Write the objective function z=Ax+By Step 3: Step 3: List restrictions (constraints) as inequalities Step 4: Solve using Linear Programming Methodolgy

6. Writing an Objective Function Example: A manufacturer produces two models of mountain bicycles. The times (in hours) required for assembling and painting each model is given: The maximum total weekly hours available are : 200 hrs for assembly and 108 hours for painting. The profits per unit are $25 for model A and $15 for model B. How many of each type should be produced to maximize profit ? Example: A manufacturer produces two models of mountain bicycles. The times (in hours) required for assembling and painting each model is given: The maximum total weekly hours available are : 200 hrs for assembly and 108 hours for painting. The profits per unit are $25 for model A and $15 for model B. How many of each type should be produced to maximize profit ? Model AModel B Assembling54 Painting23