# QMB12E Case Chapter 09 - Chapter 9 Linear Programming...

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Chapter 9 Linear Programming Applications in Marketing, Finance, and Operations Management Case Problem 1: Planning an Advertising Campaign The decision variables are as follows: T1 = number of television advertisements with rating of 90 and 4000 new customers T2 = number of television advertisements with rating of 40 and 1500 new customers R1 = number of radio advertisements with rating of 25 and 2000 new customers R2 = number of radio advertisements with rating of 15 and 1200 new customers N1 = number of newspaper advertisements with rating of 10 and 1000 new customers N2 = number of newspaper advertisements with rating of 5 and 800 new customers The Linear Programming Model and solution are as follows: MAX 90T1+55T2+25R1+20R2+10N1+5N2 S.T. 1) 1T1<=10 2) 1R1<=15 3) 1N1<=20 4) 10000T1+10000T2+3000R1+3000R2+1000N1+1000N2<=279000 5) 4000T1+1500T2+2000R1+1200R2+1000N1+800N2>=100000 6) -2T1-2T2+1R1+1R2>=0 7) 1T1+1T2<=20 8) 10000T1+10000T2>=140000 9) 3000R1+3000R2<=99000 10) 1000N1+1000N2>=30000 OPTIMAL SOLUTION Optimal Objective Value 2160.00000 Variable Value Reduced Cost T1 10.00000 0.00000 T2 5.00000 0.00000 R1 15.00000 0.00000 R2 18.00000 0.00000 N1 20.00000 0.00000 N2 10.00000 0.00000 Constraint Slack/Surplus Shadow Price CP - 34
Chapter 9 1 0.00000 35.00000 2 0.00000 5.00000 3 0.00000 5.00000 4 0.00000 0.00550 5 27100.00000 0.00000 6 3.00000 0.00000 7 5.00000 0.00000 8 10000.00000 0.00000 9 0.00000 0.00117 10 0.00000 -0.00050 Objective Allowable Allowable Coefficient Increase Decrease 90.00000 Infinite 35.00000 55.00000 11.66667 5.00000 25.00000 Infinite 5.00000 20.00000 5.00000 3.50000 10.00000 Infinite 5.00000 5.00000 0.50000 Infinite RHS Allowable Allowable Value Increase Decrease 10.00000 5.00000 10.00000 15.00000 18.00000 15.00000 20.00000 10.00000 20.00000 279000.00000 15000.00000 10000.00000 100000.00000 27100.00000 Infinite 0.00000 3.00000 Infinite 20.00000 Infinite 5.00000 140000.00000 10000.00000 Infinite 99000.00000 10000.00000 5625.00000 30000.00000 10000.00000 10000.00000 1. Summary of the Optimal Solution T1 + T2 = 10 + 5 = 15 Television advertisements R1 + R2 = 15 + 18 = 33 Radio advertisements N1 + N2 = 20 + 10 = 30 Newspaper advertisements Advertising Schedule: Media Number of Ads Budget Television 15 \$150,000 Radio 33 99,000 Newspaper 30 30,000 CP - 35
Solutions to Case Problems Totals 78 \$279,000 Total Exposure Rating: 2,160 Total New Customers Reached: 127,100 (Surplus constraint 5) 2. The shadow price shows that total exposure increases 0.0055 points for each one dollar increase in the advertising budget. Right Hand Side Ranges show this shadow price applies for a budget increase of up to \$15,000. Thus the shadow price applies for the \$10,000 increase. Total Exposure Rating would increase by 10,000(0.0055) = 55 points A \$10,000 increase in the advertising budget is a 3.6% increase. But, it only provides a 2.54% increase in total exposure. Management may decide that the additional exposure is not worth the cost. This is a discussion point. 3. The ranges for the exposure rating of 90 for the first 10 television ads show that the solution remains optimal as long as the exposure rating is 55 or higher. This indicates that the solution is not very sensitive to the exposure rating HJ has provided. Indeed, we would draw the same conclusion after reviewing the next four ranges. We could conclude that Flamingo does not have to be concerned about the exact exposure rating. The only concern might be the newspaper exposure rating of 5. A rating of 5.5 or better can be expected to alter the current optimal solution. 4. Remove constraint #5 for the linear programming model and use it to develop the objective function: MAX 4000T1+1500T2+2000R1+1200R2+1000N1+800N2 Solving provides the following Optimal Solution T1 + T2 = 10 + 4 = 14 Television advertisements