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In Example 7.5, we implied that each of the five observations was from one period of time, such as a particular week. Suppose instead that each is an average over several weeks. For example, the 4.7 million exposures corresponding to one ad might really be an average over 15 different weeks where one ad was shown in each of these weeks. Similarly, the 90.3 million exposures corresponding to 50 ads might really be an average over only three different weeks where 50 ads were shown in each of these weeks. If the observations are really averages over different numbers of weeks, then simply summing the squared prediction errors doesn’t seem appropriate. For example, it seems more appropriate that an average over 15 weeks should get five times as much weight as an average over only three weeks. Assume the five observations in the example are really averages over 15 weeks, 10 weeks, 4 weeks, 3 weeks, and 1 week, respectively. Devise an appropriate fitting function, to replace sum of squared errors or RMSE, and use it to find the best fit. EXAMPLE 7.5 E STIMATING AN A DVERTISING R ESPONSE F UNCTION Recall that the General Flakes Company from Example 4.1 of Chapter 4 sells a brand of low-fat breakfast cereal that appeals to people of all age groups and both genders. The company has advertised this product in various media for a number of years and has accumulated data on its advertising effectiveness. For example, the company has tracked the number of exposures to young men from ads placed on a particular television show for five different time periods. In each of these time periods, a different number of ads was used. Specifically, the numbers of ads were 1, 8, 20, 50, and 100. The corresponding numbers of exposures (in millions) were 4.7, 22.1, 48.7, 90.3, and 130.5. What type of nonlinear response function might fit these data well?

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Practical Management Science

6th Edition
WINSTON + 1 other
Publisher: Cengage,
ISBN: 9781337406659

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BuyFindarrow_forward

Practical Management Science

6th Edition
WINSTON + 1 other
Publisher: Cengage,
ISBN: 9781337406659
Chapter 7.4, Problem 23P
Textbook Problem
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In Example 7.5, we implied that each of the five observations was from one period of time, such as a particular week. Suppose instead that each is an average over several weeks. For example, the 4.7 million exposures corresponding to one ad might really be an average over 15 different weeks where one ad was shown in each of these weeks. Similarly, the 90.3 million exposures corresponding to 50 ads might really be an average over only three different weeks where 50 ads were shown in each of these weeks. If the observations are really averages over different numbers of weeks, then simply summing the squared prediction errors doesn’t seem appropriate. For example, it seems more appropriate that an average over 15 weeks should get five times as much weight as an average over only three weeks. Assume the five observations in the example are really averages over 15 weeks, 10 weeks, 4 weeks, 3 weeks, and 1 week, respectively. Devise an appropriate fitting function, to replace sum of squared errors or RMSE, and use it to find the best fit.

EXAMPLE 7.5 ESTIMATING AN ADVERTISING RESPONSE FUNCTION

Recall that the General Flakes Company from Example 4.1 of Chapter 4 sells a brand of low-fat breakfast cereal that appeals to people of all age groups and both genders. The company has advertised this product in various media for a number of years and has accumulated data on its advertising effectiveness. For example, the company has tracked the number of exposures to young men from ads placed on a particular television show for five different time periods. In each of these time periods, a different number of ads was used. Specifically, the numbers of ads were 1, 8, 20, 50, and 100. The corresponding numbers of exposures (in millions) were 4.7, 22.1, 48.7, 90.3, and 130.5. What type of nonlinear response function might fit these data well?

Summary Introduction

To devise: An appropriate fitting function.

Non-linear programming (NLP):

Non-linear programming (NLP) is used in complex optimization problems where the objectives or constraints or sometimes both are non-linear functions of the decision variables. A model can be termed as non-linear for more than one reason.

Explanation of Solution

Model:

Formula:

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Ch. 7.3 - ELECTRICITY PRICING AT FLORIDA POWER AND LIGHT...Ch. 7.3 - In the electricity pricing model in Example 7.4,...Ch. 7.3 - Continuing Problem 6, suppose the company is...Ch. 7.3 - PRICING SUITS AT SULLIVANS Sullivans is a retailer...Ch. 7.3 - Continuing the previous problem (the model in part...Ch. 7.4 - In estimating the advertising response function in...Ch. 7.4 - The best-fitting advertising response function in...Ch. 7.4 - In judging the fit of the estimated response...Ch. 7.4 - Starting with the solution to the advertising...Ch. 7.4 - The preceding problem indicates how fewer...Ch. 7.4 - In the solution to the advertising selection model...Ch. 7.4 - The advertising response function in Equation...Ch. 7.4 - In Example 7.5, we implied that each of the five...Ch. 7.5 - The Lafferty Company wants to locate a warehouse...Ch. 7.5 - The Lafferty Company wants to locate a warehouse...Ch. 7.5 - The Lafferty Company wants to locate a warehouse...Ch. 7.5 - The Lafferty Company wants to locate a warehouse...Ch. 7.6 - The file P07_28.xlsx lists the scores of all NFL...Ch. 7.6 - Carry out the suggestion in Modeling Issue 3. That...Ch. 7.6 - Carry out the suggestion in Modeling Issue 2 for...Ch. 7.6 - The file P07_31.xlsx contains scores on all of the...Ch. 7.6 - By the time you are reading this, the 2016 NFL...Ch. 7.6 - By the time you are reading this, the 20162017 NBA...Ch. 7.6 - The method for rating teams in Example 7.8 is...Ch. 7.7 - For each of the following, answer whether it makes...Ch. 7.7 - Add a new stock, stock 4, to the model in Example...Ch. 7.7 - In the model in Example 7.9, stock 2 is not in the...Ch. 7.7 - The stocks in Example 7.9 are all positively...Ch. 7.7 - The file P07_39.xlsx contains historical monthly...Ch. 7.7 - This problem continues using the data from the...Ch. 7.7 - Continuing the previous problem, find the...Ch. 7.7 - In many cases, the portfolio return is at least...Ch. 7.8 - Given the data in the file Stock Beta.xlsx,...Ch. 7.8 - Solve the previous problem, but analyze GE instead...Ch. 7 - Suppose Ford currently sells 250,000 Ford Mustangs...Ch. 7 - Suppose the current exchange rate is 115 yen per...Ch. 7 - Another way to derive a demand function is to...Ch. 7 - A company produces widgets at plants 1 and 2. It...Ch. 7 - If a monopolist produces q units, she can charge...Ch. 7 - It costs a company 15 to purchase an hour of labor...Ch. 7 - In the previous problem, what is the minimum-cost...Ch. 7 - The cost per day of running a hospital is 300,000...Ch. 7 - Two companies are producing widgets. It costs the...Ch. 7 - A company manufactures two products. If it charges...Ch. 7 - A brewing company has 100,000 to spend on...Ch. 7 - QH Company advertises during soap operas and...Ch. 7 - A beer company has divided Bloomington into two...Ch. 7 - A company uses raw material to produce two...Ch. 7 - JJ has given you 12 million to spend on...Ch. 7 - Based on Kolesar and Blum (1973). Suppose that a...Ch. 7 - A company has five factories. The x- and...Ch. 7 - Monroe County is trying to determine where to...Ch. 7 - Consider three investments. You are given the...Ch. 7 - You have 50,000 to invest in three stocks. Let Ri...Ch. 7 - An oil company must determine how many barrels of...Ch. 7 - Suppose that you are hiring a weather forecaster...Ch. 7 - The cost of producing x units of a product during...Ch. 7 - Four items are for sale in the Dollar Value store....Ch. 7 - Most economies have a goal of maximizing the...Ch. 7 - Each morning during rush hour, 10,000 people want...Ch. 7 - Based on Grossman and Hart (1983). A salesperson...Ch. 7 - Find the minimum perimeter rectangle having area...Ch. 7 - You are given that the two nonhypotenuse sides of...Ch. 7 - A cylindrical soda can has a volume of 20 cubic...Ch. 7 - City B is 10 miles downstream from city A. City A...Ch. 7 - You can swim two miles per hour and run six miles...Ch. 7 - A triangle has a 5-inch side and a 12-inch side....Ch. 7 - Your company is about to market a new golf club....Ch. 7 - For the product mix examples (Examples 3.1 and 3.2...Ch. 7 - For the oil blending example (Example 4.4 in...Ch. 7 - For the aggregate planning example (Example 4.3 in...Ch. 7 - Consider the sports ratings model in section 7.6....Ch. 7 - Consider the sports ratings model in section 7.6....Ch. 7 - UE is going to invest 400 million to acquire...Ch. 7 - Your family owns a large farm that can grow wheat,...Ch. 7 - GMS Stock Hedging Kate Torelli, a security analyst...Ch. 7 - GMS Stock Hedging Kate Torelli, a security analyst...Ch. 7 - GMS Stock Hedging Kate Torelli, a security analyst...Ch. 7 - GMS Stock Hedging Kate Torelli, a security analyst...

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