Question

Flair Furniture Company produces inexpensive tables and chairs. The production process for each is similar in that both require a certain number of labor hours in the carpentry department and a certain number of labor hours in the painting department. Each table takes 3 hours of carpentry work, an hour and a half of assembly and 2 hours of finishing work. Each chair requires 4 hours of carpentry, and hour and 15 minutes of assembly and 1 hour of painting. During the current month, 2,400 hours of carpentry time, 800 hours of assembly time and 1,000 hours of painting time are available. Each table sold results in a profit contribution of $7, and each chair sold yields a profit contribution of $5.

a. Set up a Solver model for determine the number of tables and chairs to produce that will maximize total profit contribution. Run Solver and generate the Answer Report and Sensitivity Report.

b. Identify the binding and nonbinding constraints and explain what it means.

c. Construct the sensitivity range for the objective function coefficients. Give an interpretation for each range.

d. Construct the sensitivity range for the right hand side coefficient of constraints. Give an interpretation for each range and the corresponding shadow price of all the constraints.

e. Suppose that 100 hours of additional labor can be added. In which department would you add these hours? Explain why. How much additional profit can be generated by this addition?

Answer 1

a.

Decision variables:

X = Number of tables

Y = Number of chairs

Objective function:

Max Z = 7X + 5Y

Constraints:

3X + 4Y 2400

1.5X + 1.25Y 800

2X + Y 1000

X, Y 0

Linear model:

Max Z = 7X + 5Y

sub.to the constraints

3X + 4Y 2400

1.5X + 1.25Y 800

2X + Y 1000

X, Y 0

Solver model:

Answer report:

Sensitivity report:

Optimal values:

Number of tables = 450

Number of chairs = 100

Total profit = 3650

b.

Constraints of assembly time and paint time are binding and the constraints of carpentry and non-negativity are not-binding. Binding constraints are the contraints which fully utilized the available resources. Non-binding constraints are the constraints which do not full utilize the available resources.

c.

For X, the objective coefficient is 7, allowable increase is 3 and allowable decrease is 1. So the sensitivity range for the objective coefficient of X is (6, 10). This is the range for which the optimal value for X does not change.

For Y, the objective coefficient is 5, allowable increase is 0.83 and allowable decrease is 1.5. So the sensitivity range for the objective coefficient of Y is (3.5, 5.83). This is the range for which the optimal value for Y does not change.

d.

For the carpentry constraint, the right hand side value is 2400 with allowable increase infinity and allowable decrease 650. So the sensitivity range for carpentry is (1750, infinity). This is the range for which the shadow price works where shadow price is the change in the objective function for one unit change in the right hand side value. The optimal values for X and Y would not change if the carpentry hours is between (1750, infinity). There would be no change in the total profit also as the shadow price is zero.

For the assembly constraint, the right hand side value is 800 with allowable increase130 and allowable decrease 50. So the sensitivity range for assembly is (750, 930). The optimal values for X and Y would not change if the assembly hours is between (750, 930). There would be change in the objective value by 3 for each unit change in the right hand side value within the sensitivity range.

For the painting constraint, the right hand side value is 1000 with allowable increase 66.67 and allowable decrease 288.89. So the sensitivity range for painting is (1066.67, 711.11). The optimal values for X and Y would not change if the painting hours is between (1066.67, 711.11). There would be change in the objective value by 1.25 for each unit change in the right hand side value within the sensitivity range.

e.

The additional 100 hours can be added to the binding constraint and with higher shadow price. Assembly constraint have the allowable increase of right hand side is above 100 and the shadow pirce is the highest (3). So adding 100 hours to the assembly hours will generate additional profit of (3 x 100 = 300)