powerpoint slides by robert f. brookercopyright (c) 2001 by harcourt, inc. all rights reserved....
TRANSCRIPT
PowerPoint Slides by Robert F. Brooker Copyright (c) 2001 by Harcourt, Inc. All rights reserved.
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 Copyright (c) 2001 by Harcourt, Inc. All rights reserved.
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 Copyright (c) 2001 by Harcourt, Inc. All rights reserved.
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 Copyright (c) 2001 by Harcourt, Inc. All rights reserved.
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 Copyright (c) 2001 by Harcourt, Inc. All rights reserved.
Production Processes
Processes Isoquants
PowerPoint Slides by Robert F. Brooker Copyright (c) 2001 by Harcourt, Inc. All rights reserved.
Production Processes
Feasible Region Optimal Solution (S)
PowerPoint Slides by Robert F. Brooker Copyright (c) 2001 by Harcourt, Inc. All rights reserved.
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 Copyright (c) 2001 by Harcourt, Inc. All rights reserved.
Profit MaximizationQuantities of Inputs
Available perTime Period
Input Product X Product Y TotalA 1 1 7B 0.5 1 5C 0 0.5 2
Quantities of InputsRequired perUnit of Output
Maximize
Subject to
(objective function)
(input A constraint)
(input B constraint)
(input C constraint)
(nonnegativity constraint)
= $30QX + $40QY
1QX + 1QY 7
0.5QX + 1QY 5
0.5QY 2
QX, QY 0
PowerPoint Slides by Robert F. Brooker Copyright (c) 2001 by Harcourt, Inc. All rights reserved.
PowerPoint Slides by Robert F. Brooker Copyright (c) 2001 by Harcourt, Inc. All rights reserved.
Profit MaximizationMultiple Optimal Solutions
New objective function has the same slopeas the feasible region at the optimum
PowerPoint Slides by Robert F. Brooker Copyright (c) 2001 by Harcourt, Inc. All rights reserved.
Profit MaximizationAlgebraic Solution
Points of Intersection Between Constraintsare Calculated to Determine the Feasible Region
PowerPoint Slides by Robert F. Brooker Copyright (c) 2001 by Harcourt, Inc. All rights reserved.
Profit MaximizationAlgebraic Solution
Profit at each point of intersection between constraintsis calculated to determine the optimal point (E)
Corner Point QX QY Profit
0 0 0 $0D 7 0 210*E 4 3 240F 2 4 220G 0 4 160
PowerPoint Slides by Robert F. Brooker Copyright (c) 2001 by Harcourt, Inc. All rights reserved.
Cost Minimization
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)
Meat (Food X) Fish (Food Y)Price per pound $2 $3
Minimum DailyRequirements
Nutrient Meat (Food X) Fish (Food Y) TotalProtein 1 1 7
Minerals 0.5 1 5Vitamins 0 0.5 2
Units of NutrientsPer Pound of
PowerPoint Slides by Robert F. Brooker Copyright (c) 2001 by Harcourt, Inc. All rights reserved.
Cost Minimization
Feasible Region Optimal Solution (E)
PowerPoint Slides by Robert F. Brooker Copyright (c) 2001 by Harcourt, Inc. All rights reserved.
Cost MinimizationAlgebraic Solution
Cost at each point of intersection between constraintsis calculated to determine the optimal point (E)
Corner Point QX QY Cost
D 14 0 $28*E 6 4 24F 2 8 28G 0 12 36
PowerPoint Slides by Robert F. Brooker Copyright (c) 2001 by Harcourt, Inc. All rights reserved.
Dual of the ProfitMaximization 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 Copyright (c) 2001 by Harcourt, Inc. All rights reserved.
Dual of the CostMinimization Problem
Maximize
Subject to
= 14VP + 10VM + 6VV
1VP + 1VM + 1VV $30
2VP + 1VM + 0.5VV $40
VP, VM, VV 0
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)