Warm-up

Sketch the region bounded by the system of inequalities:

1)

2)

632

4

0

0

yx

yx

y

x

102

8

2

0,0

yx

yx

yx

yx

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.

Optimization – find maximum (or minimum) of a function.

Non-Linear Optimization (Section 2.7). Linear Optimization (Section 10.8)

1. Optimization - Summary

Definition: Feasible region is the bounded region defined by the

constraints.

Continued… Maximize: z = 2x + yyxz 2

Objective function: Maximize:

Constraints : Subject to:

yxz 2

2

520

0

yx

yxy

x

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 MethodSolving 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 + yyxz 2

Objective function: Maximize:

Constraints : Subject to:

yxz 2

2

520

0

yx

yxy

x

5. A Linear Programming Problem

Continued… Maximize: z = 2x + yyxz 2Corner Point Value of Objective Function

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

6. ApplicationSolve a Linear Programming Problem in 2 variables

Step 1: Define variables

Step 2: Write the objective function z=Ax+By

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?

Model A Model BAssembling 5 4

Painting 2 3