Micro Economics in a Global Economy Chapter 8 Linear Programming PowerPoint Slides by Robert F. Brooker Harcourt, Inc. items and derived items copyright © 2001 by Harcourt, Inc.

Linear Programming Mathematical Technique for Solving Constrained Maximization and Minimization Problems Assumes that the Objective Function is Linear Assumes that All Constraints Are Linear PowerPoint Slides by Robert F. Brooker Harcourt, Inc. items and derived items copyright © 2001 by Harcourt, Inc.

Applications of Linear Programming Optimal Process Selection Optimal Product Mix Satisfying Minimum Product Requirements Long-Run Capacity Planning Least Cost Shipping Route (Transportation Problems) PowerPoint Slides by Robert F. Brooker Harcourt, Inc. items and derived items copyright © 2001 by Harcourt, Inc.

Applications of Linear Programming Airline Operations Planning Output Planning with Resource and Process Capacity Constraints Distribution of Advertising Budget Routing of Long-Distance Phone Calls Investment Portfolio Selection Allocation of Personnel Among Activities PowerPoint Slides by Robert F. Brooker Harcourt, Inc. items and derived items copyright © 2001 by Harcourt, Inc.

Production Processes Production processes are graphed as linear rays from the origin in input space. Production isoquants are line segments that join points of equal output on the production process rays. PowerPoint Slides by Robert F. Brooker Harcourt, Inc. items and derived items copyright © 2001 by Harcourt, Inc.

Production Processes Processes Isoquants PowerPoint Slides by Robert F. Brooker Harcourt, Inc. items and derived items copyright © 2001 by Harcourt, Inc.

Production Processes Feasible Region Optimal Solution (S) PowerPoint Slides by Robert F. Brooker Harcourt, Inc. items and derived items copyright © 2001 by Harcourt, Inc.

Formulating and Solving Linear Programming Problems Express Objective Function as an Equation and Constraints as Inequalities Graph the Inequality Constraints and Define the Feasible Region Graph the Objective Function as a Series of Isoprofit or Isocost Lines Identify the Optimal Solution PowerPoint Slides by Robert F. Brooker Harcourt, Inc. items and derived items copyright © 2001 by Harcourt, Inc.

Profit Maximization Maximize Subject to  = $30QX + $40QY (objective function) (input A constraint) (input B constraint) (input C constraint) (nonnegativity constraint) PowerPoint Slides by Robert F. Brooker Harcourt, Inc. items and derived items copyright © 2001 by Harcourt, Inc.

PowerPoint Slides by Robert F. Brooker. Harcourt, Inc PowerPoint Slides by Robert F. Brooker Harcourt, Inc. items and derived items copyright © 2001 by Harcourt, Inc.

Multiple Optimal Solutions Profit Maximization Multiple Optimal Solutions New objective function has the same slope as the feasible region at the optimum PowerPoint Slides by Robert F. Brooker Harcourt, Inc. items and derived items copyright © 2001 by Harcourt, Inc.

Profit Maximization Algebraic Solution Points of Intersection Between Constraints are Calculated to Determine the Feasible Region PowerPoint Slides by Robert F. Brooker Harcourt, Inc. items and derived items copyright © 2001 by Harcourt, Inc.

Profit Maximization Algebraic Solution Profit at each point of intersection between constraints is calculated to determine the optimal point (E) PowerPoint Slides by Robert F. Brooker Harcourt, Inc. items and derived items copyright © 2001 by Harcourt, Inc.

Cost Minimization Minimize Subject to C = $2QX + $3QY 1QX + 2QY  14 (objective function) (protein constraint) (minerals constraint) (vitamins constraint) (nonnegativity constraint) PowerPoint Slides by Robert F. Brooker Harcourt, Inc. items and derived items copyright © 2001 by Harcourt, Inc.

Cost Minimization Feasible Region Optimal Solution (E) PowerPoint Slides by Robert F. Brooker Harcourt, Inc. items and derived items copyright © 2001 by Harcourt, Inc.

Cost Minimization Algebraic Solution Cost at each point of intersection between constraints is calculated to determine the optimal point (E) PowerPoint Slides by Robert F. Brooker Harcourt, Inc. items and derived items copyright © 2001 by Harcourt, Inc.

Dual of the Profit Maximization Problem Maximize Subject to  = $30QX + $40QY 1QX + 1QY  7 0.5QX + 1QY  5 0.5QY  2 QX, QY  0 (objective function) (input A constraint) (input B constraint) (input C constraint) (nonnegativity constraint) Minimize Subject to C = 7VA + 5VB + 2VC 1VA + 0.5VB  $30 1VA + 1VB + 0.5VC  $40 VA, VB, VC  0 PowerPoint Slides by Robert F. Brooker Harcourt, Inc. items and derived items copyright © 2001 by Harcourt, Inc.

Dual of the Cost Minimization Problem Minimize Subject to C = $2QX + $3QY 1QX + 2QY  14 1QX + 1QY  10 1QX + 0.5QY  6 QX, QY  0 (objective function) (protein constraint) (minerals constraint) (vitamins constraint) (nonnegativity constraint) Maximize Subject to  = 14VP + 10VM + 6VV 1VP + 1VM + 1VV  $30 2VP + 1VM + 0.5VV  $40 VP, VM, VV  0 PowerPoint Slides by Robert F. Brooker Harcourt, Inc. items and derived items copyright © 2001 by Harcourt, Inc.