It's a lot of words, but short questions. I got most of it, but I'm confused on a and c. Help would be greatly appreciated:
Theater Seven wants to increase its popularity. They want to determine the best number of newspaper (N), Radio (R), and Television (T) ads to produce in order to maximize profits. Constraints include production capacity limitations (time available in minutes) in each of three departments (technical support, design, workforce) as well as a constraint that requires the production of at least 1000 newspaper ads. The linear programming model of The Theater Seven’s problem is
Max z = 6 N + 10 R + 8 T
s.t.
24 N +20 R + 16 T < (or equal to) 36,000 Technical support
30 N +30 R + 24 T < (or equal to) 36,000 Design
6 N + 8 R + 4 T < (or equal to) 18,000 Workforce
1 N > (or equal to) 1,000 Newspaper ads
N, R, T > ( or equal to) 0
Refer to the enclosed Excel computer solution and answer the following:
a. Overtime rates in the Design are $12 per hour. Would you recommend
that the company consider using overtime in that department?
c. Suppose that the profit contribution of the Radio ad is increased by $1. How do you expect the solution to change?
Microsoft Excel 9.0 Answer Report
Target Cell (Max)
Cell Name Original Value Final Value
$B$10 MAX 0 8000
Adjustable Cells
Cell Name Original Value Final Value
$C$9 dec. variables N 0 1000
$D$9 dec. variables R 0 200
$E$9 dec. variables T 0 0
Constraints
Cell Name Cell Value Formula Status Slack
$C$12 Tech. Spport LHS 28000 $C$12<=$E$12 Not Binding 8000
$C$13 Design LHS 36000 $C$13<=$E$13 Binding 0
$C$14 Workforce LHS 7600 $C$14<=$E$14 Not Binding 10400
$C$15 Newspaper ads LHS 1000 $C$15>=$E$15 Binding 0
Microsoft Excel 9.0 Sensitivity Report
Adjustable Cells
Final Reduced Objective Allowable Allowable
Cell Name Value Cost Coefficient Increase Decrease
$C$9 dec. variables N 1000 0 6 4 1E+30
$D$9 dec. variables R 200 0 10 1E+30 0
$E$9 dec. variables T 0 0 8 0 1E+30
Constraints
Final Shadow Constraint Allowable Allowable
Cell Name Value Price R.H. Side Increase Decrease
$C$12 Tech. Spport LHS 28000 0 36000 1E+30 8000
$C$13 Design LHS 36000 0.333333333 36000 12000 6000
$C$14 Workforce LHS 7600 0 18000 1E+30 10400
$C$15 Newspaper ads LHS 1000 -4 1000 200 1000
To answer question a, we need to analyze the Excel computer solution. Looking at the Sensitivity Report, we can see that in the "Constraints" section, the Design constraint is binding, meaning that it is limiting the solution. The shadow price for the Design constraint is 0.333, indicating that for every extra unit of Design capacity available, the objective function (profit) will increase by $0.333.
Since the overtime rate in the Design department is $12 per hour, we need to compare this cost with the potential increase in profit. If the additional profit obtained by using overtime in the Design department is higher than the cost of overtime, it would be recommended to consider using overtime in that department.
To calculate the potential increase in profit, we can multiply the shadow price (0.333) by the overtime rate ($12) and by the number of available hours for overtime. In this case, we don't have information about the number of available hours for overtime, so we can't make a definitive recommendation based on this information alone.
To answer question c, we need to analyze the Sensitivity Report again. In the "Adjustable Cells" section, we can see that the reduced cost for the Radio variable is 0. This means that the objective function (profit) is not affected by changes in the Radio variable's value.
Therefore, if the profit contribution of the Radio ad is increased by $1, it would not have an impact on the solution. The values of the Dec. Variables and the objective function (MAX) will remain the same.
In summary:
a. Based on the given information, we can't make a definitive recommendation about using overtime in the Design department without knowing the number of available hours for overtime.
c. Increasing the profit contribution of the Radio ad by $1 will not change the solution.