Managerial Economics in a
Global Economy
Chapter 8
Linear Programming
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
PowerPoint Slides by Robert F. Brooker
Isoquants
Copyright (c) 2001 by Harcourt, Inc. All rights reserved.
Production Processes
Feasible Region
PowerPoint Slides by Robert F. Brooker
Optimal Solution (S)
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 Maximization
Input
A
B
C
Quantities of Inputs
Required per
Unit of Output
Product X
Product Y
1
1
0.5
1
0
0.5
Quantities of Inputs
Available per
Time Period
Total
7
5
2
Maximize
 = $30QX + $40QY
(objective function)
Subject to
1QX + 1QY  7
(input A constraint)
0.5QX + 1QY  5
(input B constraint)
0.5QY  2
(input C constraint)
QX, QY  0
PowerPoint Slides by Robert F. Brooker
(nonnegativity constraint)
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 Maximization
Multiple Optimal Solutions
New objective function has the same slope
as the feasible region at the optimum
PowerPoint Slides by Robert F. Brooker
Copyright (c) 2001 by Harcourt, Inc. All rights reserved.
Profit Maximization
Algebraic Solution
Points of Intersection Between Constraints
are Calculated to Determine the Feasible Region
PowerPoint Slides by Robert F. Brooker
Copyright (c) 2001 by Harcourt, Inc. All rights reserved.
Profit Maximization
Algebraic Solution
Profit at each point of intersection between constraints
is calculated to determine the optimal point (E)
Corner Point
0
D
*E
F
G
PowerPoint Slides by Robert F. Brooker
QX
0
7
4
2
0
QY
0
0
3
4
4
Profit
$0
210
240
220
160
Copyright (c) 2001 by Harcourt, Inc. All rights reserved.
Cost Minimization
Price per pound
Nutrient
Protein
Minerals
Vitamins
Minimize
Subject to
Meat (Food X)
$2
Fish (Food Y)
$3
Units of Nutrients
Per Pound of
Meat (Food X)
Fish (Food Y)
1
1
0.5
1
0
0.5
Minimum Daily
Requirements
Total
7
5
2
C = $2QX + $3QY
(objective function)
1QX + 2QY  14
(protein constraint)
1QX + 1QY  10
(minerals constraint)
1QX + 0.5QY  6
(vitamins constraint)
QX, QY  0
PowerPoint Slides by Robert F. Brooker
(nonnegativity constraint)
Copyright (c) 2001 by Harcourt, Inc. All rights reserved.
Cost Minimization
Feasible Region
PowerPoint Slides by Robert F. Brooker
Optimal Solution (E)
Copyright (c) 2001 by Harcourt, Inc. All rights reserved.
Cost Minimization
Algebraic Solution
Cost at each point of intersection between constraints
is calculated to determine the optimal point (E)
Corner Point
D
*E
F
G
PowerPoint Slides by Robert F. Brooker
QX
14
6
2
0
QY
0
4
8
12
Cost
$28
24
28
36
Copyright (c) 2001 by Harcourt, Inc. All rights reserved.
Dual of the Profit
Maximization Problem
Maximize
 = $30QX + $40QY
(objective function)
Subject to
1QX + 1QY  7
(input A constraint)
0.5QX + 1QY  5
(input B constraint)
0.5QY  2
(input C constraint)
QX, QY  0
Minimize
Subject to
(nonnegativity constraint)
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 Cost
Minimization Problem
Minimize
Subject to
C = $2QX + $3QY
(objective function)
1QX + 2QY  14
(protein constraint)
1QX + 1QY  10
(minerals constraint)
1QX + 0.5QY  6
(vitamins constraint)
QX, QY  0
(nonnegativity constraint)
Maximize
 = 14VP + 10VM + 6VV
Subject to
1VP + 1VM + 1VV  $30
2VP + 1VM + 0.5VV  $40
VP, VM, VV  0
PowerPoint Slides by Robert F. Brooker
Copyright (c) 2001 by Harcourt, Inc. All rights reserved.