Part I: Maximizing The Number of Exposures of Ads Amber has recently faced quite a reduced number of customers due to the Corona pandemic. Vijay has decided to develop a promotional campaign for Amber restaurants to address this issue. Amber's dishes have a lot going for it. They have great taste, nutritious and some of the dishes such as Amber's famous Dahi Kebab" are quite crispy! Dahi Kebab is a vegetarian dish made from hung yogurt, fresh cheese, cashew, and green papaya relish. Overall, Amber's meals have the makings of a winning promotional campaign. Vijay has employed a San Francisco-based marketing consulting firm, BayCreative, to help design a promotional ad campaign to achieve Amber, the largest possible exposure to customers in Northern California during a one-year period. Amber will pay BayCreative a fee based on services performed (not to exceed $1 million) and has also allocated $4 million for advertising expenses. BayCreative has identified three effective advertising media for promoting Amber's restaurants: Medium 1: Television commercials on Bay-Area based TV stations. Medium 2: Advertisements in the social media such as Facebook and Instagram. Medium 3: Advertisements in major newspapers. The problem now is to determine which levels should be chosen for these advertising activities to obtain the highest number of ad exposures to customers. To determine the best mix of advertising activity levels, it is necessary to identify the overall performance measure for the problem and then the contribution of each activity toward this measure. Vijay's ultimate goal is to maximize Amber's profits, but making a direct connection between advertising exposure and profits is not easy. Therefore, as a proxy, Vijay decides to use the expected number of impressions as the overall measure of performance, where each displaying of an advertisement to an individual counts as one impression. By Vijay's request, BayCreative has come up with preliminary plans for advertisements in the three media. It has also estimated the expected number of impressions for each advertisement through each medium, as given in the bottom row of Table 1. Table 1: Costs and impressions data Each TV Each Social Media Commercial Ad Cost Category Each Newspaper Ad $300,000 $150,000 $100,000 Advertising costs for one year Planning costs Expected number of impressions $90,000 $30,000 $40,000 1,300,000 600,000 500,000 The total number of ads through the different media is restricted by both the allocated advertising budget (a limit of $4 million) and the planning budget (a limit of $1 million for the fee to BayCreative). Another restriction is that there are only five commercial spots available in major Bay Area TV stations during the next year (medium 1). The other two media have an ample number of spots available. Consequently, the three limited resources for this problem are . Resource 1: Advertising budget ($4 million), Resource 2: Planning budget ($1 million), Resource 3: TV commercial spots available (5). Table 1 shows how much of the advertising budget and the planning budget would be used by each advertisement in the respective media: The first row gives the cost per ad in each medium. (The cost of using only a fraction of an advertising spot is assumed to be that fraction of the cost given in the table.) The second row shows Bay Creative's estimates of its total cost (including overhead and profit) to design and develop each advertisement for the respective media. The last row then gives the expected number of impressions per advertisement. Since the promotional campaign is for Amber's award-winning restaurants with distinctive dishes and a modem environment, they have special appeal to the young people segment (whose ages are from 18 to less than 30 years old) and the middle-aged people segment (whose ages are from 30 to less than 55 years old). Vijay beleives that these two distinct customer segments should be targeted separately. Consequently, he has made two requirements for his campaign: Requirement 1: During the next year, the ads of one type or another should be displayed to at least 5 million Bay Area residents in the young people segment. Requirement 2: During the next year, the ads of one type or another should be displayed to at least 5 million Bay Area residents in the middle-aged people segment. In effect, these two requirements are minimum acceptable levels for two special benefits to be achieved by the advertising activities. Benefit 1: Promoting Amber to the young people segment. Benefit 2: Promoting Amber to the middle-aged people segment. Because of the way the requirements have been articulated, the level of each of these benefits is measured by the number of people in the specified category that are reached by the advertising To enable the construction of the corresponding benefit constraints, Vijay asks BayCreative to estimate how much the advertisement in each of the media will contribute to each benefit, as measured by the number of people reached in the specified category. These estimates are given in Table 2. Table 2: Number reached in each audience segment Each TV Each Social Media Each Newspaper Minimum Acceptable Commercial Ad Level Audience Segment Ad Young People 1.2 million 0.2 million O million 5 million Middle Aged People 0.5 million 0.2 million 0.2 million 5 million Vijay has one more consideration that he wants to incorporate into the model. He is a strong believer in the use of discount coupons. So, He always allocates a significant portion of Amber's annual marketing budget to offer these coupons. Vijay has $ 1,490,000 left from this year's allocation for discount coupons and has decided to use all of this budget to offer discount coupons. Both medium 2 (advertisements in the social media like Facebook and Instagram) and medium 3 (advertisements in major newspapers) will feature discount coupons. The estimates of the amount of coupon redemption per ad in each of these media are given in Table 3. Table 3: Contribution toward the required amount Requirement Each TV Each Social Media Each Newspaper Madmum acceptable Commercial Level Ad Ad Coupon Redemption $0 $40,000 $120,000 $1,490,000 Vijay has asked for help from your team! for the planning of his promotional campaign. YOUR MISSION! Determine the decision variables Formulate the optimization model with all the constraints Coding in R, find the optimal number of advertisements to run in each media that maximize the expected number of exposures while satisfying all the constraints. What is the optimal profit value? What are the optimal values for the variables? The advertising campaign's ultimate goal is to maximize Amber's profits from sales. However, it is not easy to make a direct connection between advertising impressions and profits. Therefore, Vijay chose the expected number of impressions as a proxy for profit in Part I. Having done this, Vijay feels uncomfortable. He realizes that the total profit may not be proportional to the total number of impressions of ads. So, his approximation might not be accuate enough. The most important reason for it is that running too many ads in any of the media passes a saturation level beyond which the effect of one more ad in convincing a customer to visit Amber is substantially reduced. Nevertheless, when the objective function is the expected number of impressions, having an individual see the ad one more time after being saturated adds the same amount to the objective function as seeing the ad for the first time. To check the results obtained in Part I, Vijay now decides to incorporate Amber's actual profit as the overall performance measure directly into the objective function, instead of using any approximation as in Part I. To do so, Vijay carefully defines the profit function as the total profit obtained from the number of first-time customer visits for dinner/lunch because of the ad campaign. This excludes profits from impulse visits by customers who have seen no ads, since these visits have no relevance for evaluating the ad campaign. Repeat visits are also excluded from consideration because these depend mainly on customers' reactions to their first experience at Amber instead of the ad campaign. Vijay asks BayCreative to develop estimates of the number of first-time customer visits that should result from various numbers of ads in each of the media. BayCreative estimates based on some past data are given below. The estimated relationship between the number of TV commercials (TV) on the number of first-time customer visits (millions) (S) (either lunch or dinner) per year to all Amber restaurants: S = -0.17 v2 + 1.13TV -0.04 The estimated relationship between the number of the social media ads (M) on the number of first-time customer visits (millions) (S) (either lunch or dinner) per year to all Amber restaurants: S = -0.002M2 +0.124M + 0.14 The estimated relationship between the number of newspaper ads (N) on the number of first-time customer visits (millions) (S) (either lunch or dinner) per year to all Amber restaurants: S = -0.0321N2 +0.706N - 0.09 BayCreative also reports that it is reasonable to assume that the sales that result from advertising in one of the media are not affected by the amount of advertising in the other media since the audiences for the different media are usually different. It is estimated that the company's profit from serving each customer (either for lunch or dinner) will be, on average, $5 per customer. This profit includes the price of dishes charged minus the various costs for preparing the dish and the service. However, this profit excludes the advertising costs and planning costs specified BayCreative also reports that it is reasonable to assume that the sales that result from advertising in one of the media are not affected by the amount of advertising in the other media since the audiences for the different media are usually different. It is estimated that the company's profit from serving each customer (either for lunch or dinner) will be, on average, $5 per customer. This profit includes the price of dishes charged minus the various costs for preparing the dish and the service. However, this profit excludes the advertising costs and planning costs specified in Part I for the advertising campaign. Therefore, Vijay wants to include these costs in his definition of the total profit that should be considered for determining the best advertising mix. Vijay has asked again for your consulting teams help! YOUR MISSION! Using your results from Part 1, formulate the total profit function (as defined by Vijay). Use R (or any other language) to determine the optimal number of ads to run in each medium to maximize the total profit while satisfying all the constraints in Part I. What is the optimal profit value? What are the optimal values for the variables? Based on your calculation, what can you conclude about the accuracy of Vjay's approximation in Part I (i.e., finding the optimal variables by using the expected number of impressions, rather than maximizing the actual profit)? Part I: Maximizing The Number of Exposures of Ads Amber has recently faced quite a reduced number of customers due to the Corona pandemic. Vijay has decided to develop a promotional campaign for Amber restaurants to address this issue. Amber's dishes have a lot going for it. They have great taste, nutritious and some of the dishes such as Amber's famous Dahi Kebab" are quite crispy! Dahi Kebab is a vegetarian dish made from hung yogurt, fresh cheese, cashew, and green papaya relish. Overall, Amber's meals have the makings of a winning promotional campaign. Vijay has employed a San Francisco-based marketing consulting firm, BayCreative, to help design a promotional ad campaign to achieve Amber, the largest possible exposure to customers in Northern California during a one-year period. Amber will pay BayCreative a fee based on services performed (not to exceed $1 million) and has also allocated $4 million for advertising expenses. BayCreative has identified three effective advertising media for promoting Amber's restaurants: Medium 1: Television commercials on Bay-Area based TV stations. Medium 2: Advertisements in the social media such as Facebook and Instagram. Medium 3: Advertisements in major newspapers. The problem now is to determine which levels should be chosen for these advertising activities to obtain the highest number of ad exposures to customers. To determine the best mix of advertising activity levels, it is necessary to identify the overall performance measure for the problem and then the contribution of each activity toward this measure. Vijay's ultimate goal is to maximize Amber's profits, but making a direct connection between advertising exposure and profits is not easy. Therefore, as a proxy, Vijay decides to use the expected number of impressions as the overall measure of performance, where each displaying of an advertisement to an individual counts as one impression. By Vijay's request, BayCreative has come up with preliminary plans for advertisements in the three media. It has also estimated the expected number of impressions for each advertisement through each medium, as given in the bottom row of Table 1. Table 1: Costs and impressions data Each TV Each Social Media Commercial Ad Cost Category Each Newspaper Ad $300,000 $150,000 $100,000 Advertising costs for one year Planning costs Expected number of impressions $90,000 $30,000 $40,000 1,300,000 600,000 500,000 The total number of ads through the different media is restricted by both the allocated advertising budget (a limit of $4 million) and the planning budget (a limit of $1 million for the fee to BayCreative). Another restriction is that there are only five commercial spots available in major Bay Area TV stations during the next year (medium 1). The other two media have an ample number of spots available. Consequently, the three limited resources for this problem are . Resource 1: Advertising budget ($4 million), Resource 2: Planning budget ($1 million), Resource 3: TV commercial spots available (5). Table 1 shows how much of the advertising budget and the planning budget would be used by each advertisement in the respective media: The first row gives the cost per ad in each medium. (The cost of using only a fraction of an advertising spot is assumed to be that fraction of the cost given in the table.) The second row shows Bay Creative's estimates of its total cost (including overhead and profit) to design and develop each advertisement for the respective media. The last row then gives the expected number of impressions per advertisement. Since the promotional campaign is for Amber's award-winning restaurants with distinctive dishes and a modem environment, they have special appeal to the young people segment (whose ages are from 18 to less than 30 years old) and the middle-aged people segment (whose ages are from 30 to less than 55 years old). Vijay beleives that these two distinct customer segments should be targeted separately. Consequently, he has made two requirements for his campaign: Requirement 1: During the next year, the ads of one type or another should be displayed to at least 5 million Bay Area residents in the young people segment. Requirement 2: During the next year, the ads of one type or another should be displayed to at least 5 million Bay Area residents in the middle-aged people segment. In effect, these two requirements are minimum acceptable levels for two special benefits to be achieved by the advertising activities. Benefit 1: Promoting Amber to the young people segment. Benefit 2: Promoting Amber to the middle-aged people segment. Because of the way the requirements have been articulated, the level of each of these benefits is measured by the number of people in the specified category that are reached by the advertising To enable the construction of the corresponding benefit constraints, Vijay asks BayCreative to estimate how much the advertisement in each of the media will contribute to each benefit, as measured by the number of people reached in the specified category. These estimates are given in Table 2. Table 2: Number reached in each audience segment Each TV Each Social Media Each Newspaper Minimum Acceptable Commercial Ad Level Audience Segment Ad Young People 1.2 million 0.2 million O million 5 million Middle Aged People 0.5 million 0.2 million 0.2 million 5 million Vijay has one more consideration that he wants to incorporate into the model. He is a strong believer in the use of discount coupons. So, He always allocates a significant portion of Amber's annual marketing budget to offer these coupons. Vijay has $ 1,490,000 left from this year's allocation for discount coupons and has decided to use all of this budget to offer discount coupons. Both medium 2 (advertisements in the social media like Facebook and Instagram) and medium 3 (advertisements in major newspapers) will feature discount coupons. The estimates of the amount of coupon redemption per ad in each of these media are given in Table 3. Table 3: Contribution toward the required amount Requirement Each TV Each Social Media Each Newspaper Madmum acceptable Commercial Level Ad Ad Coupon Redemption $0 $40,000 $120,000 $1,490,000 Vijay has asked for help from your team! for the planning of his promotional campaign. YOUR MISSION! Determine the decision variables Formulate the optimization model with all the constraints Coding in R, find the optimal number of advertisements to run in each media that maximize the expected number of exposures while satisfying all the constraints. What is the optimal profit value? What are the optimal values for the variables? The advertising campaign's ultimate goal is to maximize Amber's profits from sales. However, it is not easy to make a direct connection between advertising impressions and profits. Therefore, Vijay chose the expected number of impressions as a proxy for profit in Part I. Having done this, Vijay feels uncomfortable. He realizes that the total profit may not be proportional to the total number of impressions of ads. So, his approximation might not be accuate enough. The most important reason for it is that running too many ads in any of the media passes a saturation level beyond which the effect of one more ad in convincing a customer to visit Amber is substantially reduced. Nevertheless, when the objective function is the expected number of impressions, having an individual see the ad one more time after being saturated adds the same amount to the objective function as seeing the ad for the first time. To check the results obtained in Part I, Vijay now decides to incorporate Amber's actual profit as the overall performance measure directly into the objective function, instead of using any approximation as in Part I. To do so, Vijay carefully defines the profit function as the total profit obtained from the number of first-time customer visits for dinner/lunch because of the ad campaign. This excludes profits from impulse visits by customers who have seen no ads, since these visits have no relevance for evaluating the ad campaign. Repeat visits are also excluded from consideration because these depend mainly on customers' reactions to their first experience at Amber instead of the ad campaign. Vijay asks BayCreative to develop estimates of the number of first-time customer visits that should result from various numbers of ads in each of the media. BayCreative estimates based on some past data are given below. The estimated relationship between the number of TV commercials (TV) on the number of first-time customer visits (millions) (S) (either lunch or dinner) per year to all Amber restaurants: S = -0.17 v2 + 1.13TV -0.04 The estimated relationship between the number of the social media ads (M) on the number of first-time customer visits (millions) (S) (either lunch or dinner) per year to all Amber restaurants: S = -0.002M2 +0.124M + 0.14 The estimated relationship between the number of newspaper ads (N) on the number of first-time customer visits (millions) (S) (either lunch or dinner) per year to all Amber restaurants: S = -0.0321N2 +0.706N - 0.09 BayCreative also reports that it is reasonable to assume that the sales that result from advertising in one of the media are not affected by the amount of advertising in the other media since the audiences for the different media are usually different. It is estimated that the company's profit from serving each customer (either for lunch or dinner) will be, on average, $5 per customer. This profit includes the price of dishes charged minus the various costs for preparing the dish and the service. However, this profit excludes the advertising costs and planning costs specified BayCreative also reports that it is reasonable to assume that the sales that result from advertising in one of the media are not affected by the amount of advertising in the other media since the audiences for the different media are usually different. It is estimated that the company's profit from serving each customer (either for lunch or dinner) will be, on average, $5 per customer. This profit includes the price of dishes charged minus the various costs for preparing the dish and the service. However, this profit excludes the advertising costs and planning costs specified in Part I for the advertising campaign. Therefore, Vijay wants to include these costs in his definition of the total profit that should be considered for determining the best advertising mix. Vijay has asked again for your consulting teams help! YOUR MISSION! Using your results from Part 1, formulate the total profit function (as defined by Vijay). Use R (or any other language) to determine the optimal number of ads to run in each medium to maximize the total profit while satisfying all the constraints in Part I. What is the optimal profit value? What are the optimal values for the variables? Based on your calculation, what can you conclude about the accuracy of Vjay's approximation in Part I (i.e., finding the optimal variables by using the expected number of impressions, rather than maximizing the actual profit)