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Warm-up

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

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Warm-up

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

  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.

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

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

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

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

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

  8. Continued… Maximize: z = 2x + y

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

  10. 6. Application Solve 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

  11. 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 maximumtotal weekly hours available are: 200 hrs for assembly and 108 hours for painting. The profitsper unit are $25 for model A and $15 for model B. How many of each type should be produced to maximize profit?

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