# Application Problems

Harry and Melissa Jacobson produce handcrafted furniture in a workshop on their farm. They have obtained a load of 600 board feet of birch from a neighbor and are planning to produce round kitchen tables and ladder-back chairs during the next 3 months. Each table will require 30 hours of labor, each chair will require 18 hours, and between them they have a total of 480 hours of labor available. A table requires 40 board feet of wood to make, and a chair requires 15 board feet. A table earns the couple \$575 in profit, and a chair earns \$120 in profit. Most people who buy a table also want four chairs to go with it, so for every table that is produced, at least four chairs must also be made, although additional chairs can also be sold separately.

Formulate and solve an integer programming model to determine the number of tables and chairs the Jacobson’s should make to maximize profit.

Solution: Letandbe the number of tables and chairs produced, respectively. We have the following restrictions:

• Labor:

• Material:

• At least 4 chairs per table:

• Objective Function:

The problem we need to solve is therefore:

Using Excel’s solver we obtain the following results:

 4 20 Restrictions Coefficients 575 120 0 -140 Z 4700 -4
 Microsoft Excel 10.0 Answer Report Worksheet: [Eileen 2 - LP.xls]Question 1 Report Created: 8/7/2006 6:50:04 PM Target Cell (Max) Cell Name Original Value Final Value \$C\$7 Z X_T 4700 4700 Adjustable Cells Cell Name Original Value Final Value \$C\$3 X_T 4 4 \$D\$3 X_C 20 20 Constraints Cell Name Cell Value Formula Status Slack \$H\$5 Coefficients 0 \$H\$5<=0 Binding 0 \$H\$6 -140 \$H\$6<=0 Not Binding 140 \$H\$7 Z -4 \$H\$7<=0 Not Binding 4 \$C\$3 X_T 4 \$C\$3=integer Binding 0 \$D\$3 X_C 20 \$D\$3=integer Binding 0