Chapter 4
Linear Programming Applications in
Marketing, Finance and Operations
Management
Learning Objectives
1. Learn about applications of linear programming that have been encountered in practice.
2. Develop an appreciation for the diversity of problems that can be modeled as linear programs.
3. Obtain practice and experience in formulating realistic linear programming models.
4. Understand linear programming applications such as:
media selection production scheduling
portfolio selection work force assignments
blending problems
Note to Instructor
The application problems of Chapter 4 have been designed to give the student an understanding and
appreciation of the broad range of problems that can be approached by linear programming. While the
problems are indicative of the many linear programming applications, they have been kept relatively small
in order to ease the student's formulation and solution effort. Each problem will give the student an
opportunity to practice formulating a linear programming model. However, the solution and the
interpretation of the solution will require the use of a software package such as Microsoft Excel's Solver or
LINGO.
4-1
,Chapter 4
Solutions:
1. a. Let T = number of television spot advertisements
R = number of radio advertisements
N = number of newspaper advertisements
Max 100,000T + 18,000R + 40,000N
s.t.
2,000T + 300R + 600N 18,200 Budget
T 10 Max TV
R 20 Max Radio
N 10 Max News
-0.5T + 0.5R - 0.5N 0 Max 50% Radio
0.9T - 0.1R - 0.1N 0 Min 10% TV
T, R, N, 0
Budget $
Solution: T=4 $8,000
R = 14 4,200
N = 10 6,000
$18,200 Audience = 1,052,000.
OPTIMAL SOLUTION
Optimal Objective Value
1052000.00000
Variable Value Reduced Cost
T 4.00000 0.00000
R 14.00000 0.00000
N 10.00000 11826.08695
Constraint Slack/Surplus Dual Value
1 0.00000 51.30430
2 6.00000 0.00000
3 6.00000 0.00000
4 0.00000 11826.08695
5 0.00000 5217.39130
6 1.20000 0.00000
4-2
, Linear Programming Applications in Marketing, Finance and Operations Management
Objective Allowable Allowable
Coefficient Increase Decrease
100000.00000 20000.00000 118000.00000
18000.00000 Infinite 3000.00000
40000.00000 Infinite 11826.08695
RHS Allowable Allowable
Value Increase Decrease
18200.00000 13800.00000 3450.00000
10.00000 Infinite 6.00000
20.00000 Infinite 6.00000
10.00000 2.93600 10.00000
0.00000 2.93617 8.05000
0.00000 1.20000 Infinite
b. The dual value for the budget constraint is 51.30. Thus, a $100 increase in budget should provide
an increase in audience coverage of approximately 5,130. The right-hand-side range for the
budget constraint will show this interpretation is correct.
2. a. Let x1 = units of product 1 produced
x2 = units of product 2 produced
Max 30x1 + 15x2
s.t.
x1 + 0.35x2 100 Dept. A
0.30x1 + 0.20x2 36 Dept. B
0.20x1 + 0.50x2 50 Dept. C
x1, x2 0
Solution: x1 = 77.89, x2 = 63.16 Profit = 3284.21
b. The dual value for Dept. A is $15.79, for Dept. B it is $47.37, and for Dept. C it is $0.00.
Therefore we would attempt to schedule overtime in Departments A and B. Assuming the current
labor available is a sunk cost, we should be willing to pay up to $15.79 per hour in Department A
and up to $47.37 in Department B.
c. Let xA = hours of overtime in Dept. A
xB = hours of overtime in Dept. B
xC = hours of overtime in Dept. C
Max 30x1 + 15x2 - 18xA - 22.5xB - 12xC
4-3
Linear Programming Applications in
Marketing, Finance and Operations
Management
Learning Objectives
1. Learn about applications of linear programming that have been encountered in practice.
2. Develop an appreciation for the diversity of problems that can be modeled as linear programs.
3. Obtain practice and experience in formulating realistic linear programming models.
4. Understand linear programming applications such as:
media selection production scheduling
portfolio selection work force assignments
blending problems
Note to Instructor
The application problems of Chapter 4 have been designed to give the student an understanding and
appreciation of the broad range of problems that can be approached by linear programming. While the
problems are indicative of the many linear programming applications, they have been kept relatively small
in order to ease the student's formulation and solution effort. Each problem will give the student an
opportunity to practice formulating a linear programming model. However, the solution and the
interpretation of the solution will require the use of a software package such as Microsoft Excel's Solver or
LINGO.
4-1
,Chapter 4
Solutions:
1. a. Let T = number of television spot advertisements
R = number of radio advertisements
N = number of newspaper advertisements
Max 100,000T + 18,000R + 40,000N
s.t.
2,000T + 300R + 600N 18,200 Budget
T 10 Max TV
R 20 Max Radio
N 10 Max News
-0.5T + 0.5R - 0.5N 0 Max 50% Radio
0.9T - 0.1R - 0.1N 0 Min 10% TV
T, R, N, 0
Budget $
Solution: T=4 $8,000
R = 14 4,200
N = 10 6,000
$18,200 Audience = 1,052,000.
OPTIMAL SOLUTION
Optimal Objective Value
1052000.00000
Variable Value Reduced Cost
T 4.00000 0.00000
R 14.00000 0.00000
N 10.00000 11826.08695
Constraint Slack/Surplus Dual Value
1 0.00000 51.30430
2 6.00000 0.00000
3 6.00000 0.00000
4 0.00000 11826.08695
5 0.00000 5217.39130
6 1.20000 0.00000
4-2
, Linear Programming Applications in Marketing, Finance and Operations Management
Objective Allowable Allowable
Coefficient Increase Decrease
100000.00000 20000.00000 118000.00000
18000.00000 Infinite 3000.00000
40000.00000 Infinite 11826.08695
RHS Allowable Allowable
Value Increase Decrease
18200.00000 13800.00000 3450.00000
10.00000 Infinite 6.00000
20.00000 Infinite 6.00000
10.00000 2.93600 10.00000
0.00000 2.93617 8.05000
0.00000 1.20000 Infinite
b. The dual value for the budget constraint is 51.30. Thus, a $100 increase in budget should provide
an increase in audience coverage of approximately 5,130. The right-hand-side range for the
budget constraint will show this interpretation is correct.
2. a. Let x1 = units of product 1 produced
x2 = units of product 2 produced
Max 30x1 + 15x2
s.t.
x1 + 0.35x2 100 Dept. A
0.30x1 + 0.20x2 36 Dept. B
0.20x1 + 0.50x2 50 Dept. C
x1, x2 0
Solution: x1 = 77.89, x2 = 63.16 Profit = 3284.21
b. The dual value for Dept. A is $15.79, for Dept. B it is $47.37, and for Dept. C it is $0.00.
Therefore we would attempt to schedule overtime in Departments A and B. Assuming the current
labor available is a sunk cost, we should be willing to pay up to $15.79 per hour in Department A
and up to $47.37 in Department B.
c. Let xA = hours of overtime in Dept. A
xB = hours of overtime in Dept. B
xC = hours of overtime in Dept. C
Max 30x1 + 15x2 - 18xA - 22.5xB - 12xC
4-3
