XI TP1 - Number of prime-time spots to be aired on TV Channel 1; X2 T1 - Number of non prime-time spots to be aired on TV Channel 1; X3 TP2 - Number of prime-time spots to be aired on TV Channel 2; X4 T2 - Number of non prime-time spots to be aired on TV Channel 2: X5 RI - Number of spots to be aired on the radio channel; X6 G1 - Number of ads to be placed in Newspaper 1; X7 G2 - Number of ads to be placed in Newspaper 2; X8 MI - Number of ads to be placed in Magazine 1; X9 M2 - Number of ads to be placed in Magazine 2. The Objective (in units of 1000 persons): Max 5000 X1+ 3000 X2+ 4000 X3+ 2500 X4+ 400 X5+ 2300 X6+ 1300 X7+ 500 X8+ 600 X9 The Constraints: 3.0 X1+ 2.0 X2+ 2.7 X3+ 1.8 X4+0.5 X5+ 1.6 X6 + 1.2 X7+ 0.75 X8+ 0.7 X9 5 50 75 X1 + 60 X2 +100 X3+ 75 X4 + 24X5 + 50 X6 + 49 X7 + 60 X8 + 55 X9 2000 3.0 X1+ 2.0 X2+ 2.7 X3+ 1.8 X4 1.6 X6 + 1.2 X7+ 0.75 X8+0.7 X9 s 20 X1 + X2 X3 + X4 X6 X7 23 s3 X9 S3 X1, X2, X3, X4, X5, X6, X7, X8. X9 20 s 30 23 X8 The Advertising Planning Problem (Question 1) A company is planning an advertising campaign for the marketing of a new product. They y Time left the maximum number of potential customers while keeping the overall budget of the $50,000. The company's advertising agency has suggested the employment of two TV Channels, two daily newspapers, two weekly magazines and one radio station during this campaign. It has also emphasized the additional importance of reaching high-income potential customers. The advertising agency has estimated the cost of each ad, the number of potential customers and the number of high-income potential customers it will reach, as follows. In TV ads a 30 second spot is considered as a one unit, and prime-time (PT) and non prime time (non PT) ads are treated separately, since they differ both in cost and in reachability. The company would like to reach at least 2,000,000 high-income potential customers during this campaign. They would also like to spend at most $30,000 on TV ads, and at most $20,000 on printed media. Furthermore, the manager would like to give at least three ads to each TV Channel and each newspaper to keep up the good relations with these organizations. Since, the duration of the campaign is three weeks, there is time to place at most 3 ads in each of the magazines. Cost # of Potential # of High Income Media Type per Ad Cust. Reached Potential Cust. Reached TV I (PT) $ 3,000 5,000,000 75,000 TV 1 (non PT) $ 2.000 3,000,000 60,000 TV 2 (PT) $ 2,700 4,000,000 100,000 TV 2 (non PT) $ 1.800 2,500,000 75.000 Radio $ 500 400.000 24.000 Paper 1 $ 1.600 2.300,000 50,000 Paper 2 $ 1.200 1.300.000 49,000 Magazine 1 $ 750 500,000 60,000 Magazine 2 $ 700 600,000 55,000 An L.P. formulation, aimed to establish the maximum reach advertising policy, while satisfying all stated conditions is as follows. The Decision Variables: XI TP1 - Number of prime-time spots to be aired on TV Channel 1; X2 T1 - Number of non prime-time spots to be aired on TV Channel 1; X3 TP2 - Number of prime-time spots to be aired on TV Channel 2; X4 T2. Number of non nrime time snots to be aired on TV Channel 7. 1. (25 pts.) Answer the following questions regarding the advertising planning problem given on page 3. Suppose non-integer solutions are acceptable, and the following optimal solution is provided. Treat each case independently; be careful with your units. Optimal Value: 67,968,000 Optimal Primal Solution: x = (x1, x2, x3, x4, X5, X6, x7, x8, x9) = (1.8, 1.2, 0, 12.3, 0, 7.5, 3, 3, 3) Optimal Reduced Costs: r = (r1, 12, 13, 14, 15, 16, 17, rg, 19) = (0, 0, 37, 0, 127, 0, 0, 0, 0) Optimal Dual Solution: w = (w, W2, W3, W4, W5, W6, W7, wg, W9, W10) =( 2, -23, 0, 0, -313, 0, 0, -162, 260, 353) a) What is the cost (in terms of number of potential customers to be reached) to the company of insisting on at least 3 TV spots per TV Channel? Briefly explain. b) The company insists on having radio commercials within the framework of this campaign. If the budget and other conditions remain the same, how much should the number of potential customers reached be expected to increase (or decrease) as a result of an additional radio commercial? c) What is the effect on the number of ordinary potential customers reached, of the effort to reach an additional high income group potential customer (assuming that the budget and other conditions remain the same)? d) What would be the effect on the number of ordinary potential customers reached, if there was time to place up to 4 ads (instead of 3) in each of the magazines (assuming that the budget and other conditions remain the same)? e) How would you include the following condition to the model formulation? The company is negotiating a special package with TV Channel 2. So, they will either give 3.8 or 14 TV spots to this TV Channel within the framework of this campaign