Question

The situation. Advertising firm Night & Day, Inc. (NDI) has been hired to conduct an ad campaign for a client. After considering all the possible advertising media and the target market,

The situation. Advertising firm Night & Day, Inc. (NDI) has been hired to conduct an ad campaign for a client. After considering all the possible advertising media and the target market, NDI has limited next month's ads to 5 sources (listed in the table below), each of which reaches a different number of potential customers, costs a certain amount, and has its own "exposure quality." Exposure quality units measure the relative value of one ad in each source and take into account factors such as audience demographics, image presented, and ad quality Potential Customers Cost per Maximum Times Advertising Source Daytime TV Exposure Ad (S)Available per Month Quality Units 1,500 Reached per Ad 1,000 2,000 1,500 2,500 15 65 (1-min. commercial) Evening TV 3,000 10 90 (30-sec. commercial) Daily Newspaper 400 25 40 (full page ad) Sunday Newspaper 1,000 60 (1/2 page color) Radio 300 100 30 20 (30 second spot) NDI's goal is to maximize the total exposure quality from the ad campaign. The client has given NDI an advertising budget of $30,000 - see constraint [1] - and has imposed several restrictions on its use. Specifically, at least 10 TV commercials must be used, but no more than $18,000 may be spent on these commercials - see constraints [2-3]. Also, the client wants at least 50,000 potential customers to be reached through all sources - see constraint [4]. Finally, the fourth column of the table above gives limits on the number of times each advertising source may be used in a month - see constraints [5-9] LP Formulation: NDI has 5 decisions to make: . DT number of Daytime TV ads to run . ET number of Evening TV ads to run . DN number of Daily Newspaper ads to run SN number of Sunday Newspaper ads to run RAnumber of Radio Ads to run Maximize 65DT + 90ET 40DN60SN 20RA subject to: 1,500DT3,000ET 400DN 1,000SN 100RA S 30,000 DT ET 1,500DT3,000ET S 18,000 1,000DT + 2,000?? + 1,500DN + 2,500SN + 300RA 50,000 S15 10 25 [4] DT ET DN SN RA 30 DT 2 0 ET 2 0 DN 20, SN 2 0, RA 0 [10-14]

Answer 1

1a) The optimal values of the varibles are as follows:
      DT*= 10
      ET*= 0
      DN*= 25
      SN*=2
      RA*=30

1b) Total Exposure quality units:
10*65+25*40+2*60+20*30 = 2370

1c) Potential customers:
10*1000+25*1500+2*2500+30*300=99100  

In the above 3 parts we have taken the optimal values and then multiplied with the required quantity from the table.

2) As you can see in the Sensitivity Analysis, the the max coefficient of DT is 90, which means that if we change the coeffient of DT till 90 there will be no change in the optimal value.
Since in the question DT's coefficient is increased till 85 (<90) Therefore there will be no change in the Optimal value.
Ans is NO (the optimal advertising strategy will not change)

3) Using the Sensitivity Analysis table, value of SN (Sunday Newspaper) should be between 43.33 to 100, So reduction of the SN coefficient to 35 will lead to change in the optimal value.
Ans is YES (the optiaml advertising strategy will change)

4) Changing the total TV adds from 10 to 11 will lead to change of the optimal total exposure quality units value from 2370 to 2345.
Solution: In the optiamal Table we get the coefficient of S2 as 25. Here S2 is the slack variable used in the 2nd constraint. So if we increase the RHS of 2nd contraint fron 10 to 11 we are forcing S2 to take value to be 1, therefore our z value changes from 2370 to 2345 (2370-25=2345).

5) Changing the budget from $30000 to $32000 will lead to change of the optimal total exposure quality units value from 2370 to 2490.

x3 x4 Enter <, >, or- R.H.S Var. Name Maximize Constr 1 Constr 2 Constr 3 Constr 4 Constr 5 Constr 6 Constr 7 Constr 8 Constr 9 Lower Bound Upper Bound Unrestr'd (y/n)? DT 65.00 1500.00 1.00 1500.00 1000.00 1.00 0.00 0.00 0.00 0.00 0.00 infinity ET 90.00 3000.00 1.00 3000.00 2000.00 0.00 1.00 0.00 0.00 0.00 0.00 infinity DN 40.00 4000.00 0.00 0.00 1500.00 0.00 0.00 1.00 0.00 0.00 0.00 infinity SN 60.00 1000.00 0.00 0.00 2500.00 0.00 0.00 0.00 1.00 0.00 0.00 infinity RA 20.00 100.00 0.00 0.00 300.00 0.00 0.00 0.00 0.00 1.00 0.00 infinity 30000.00 10.00 18000.00 50000.00 15.00 10.00 25.00 4.00 30.00

Variable x1: DT x2: ET x3: DN x4: SN x5: RA Constraint Obj Val Contrib 650.00 0.00 1000.00 120.00 600.00 Value 10.00 0.00 25.00 2.00 30.00 RHS 30000.00 10.00 18000.00 50000.00 15.00 10.00 25.00 4.00 30.00 Obj Coeff 65.00 90.00 40.00 60.00 20.00 Slack-/Surplus+ 0.00 0.00 3000.00- 11500.00+ 2 (P) 5 (4) 5.00 10.00 0.00 2.00 0.00 9 (4) **Sensitivity Analysis*** Variable x1: DT x2: ET x3: DN x4: SN x5: RA Current Obj Coeff 65.00 90.00 40.00 60.00 20.00 Min Obj Coeff 0.00 infinity 24.00 43.33 6.00 Max Obj Coeff 90.00 155.00 infinity 100.00 infinity Reduced Cost 0.00 65.00 0.00 0.00 0.00

Phase 2 (Ilter 12 DT Basic z [max) RA SK 0.00 0.00 65.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 25.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.06 0.00 0.00 0.00 0.00 0.00 0.00 0.00 2.50 0.00 blocked 0.00 1.00 1.50 0.00 1.00 0.00 1500.00 2750.00 1.50 x1 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 infinity 0. 0. 0. 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 infinity 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 infinity 0.00 0.00 0.00 0.00 0.00 0.00 0.00 infinity SX sx13 sx10 0.00 1500.00 2750.00 0. 1500.00 2750.00 SX 0.00 Lower Bound 0.00 infinity Upper Bound Unrestr'd [y/n)?