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3. A company will soon be introducing a new product into a very competitive market and is currently planning its marketing strategy. The decision has been made to introduce the product in three phases. Phase 1 will feature making a special introductory offer of the product to the public at a greatly reduced price to attract first-time buyers. Phase 2 will involve an intensive advertising campaign to persuade these first-time buyers to continue purchasing the product at a regular price. It is known that another company will be introducing a new competitive product at about the time that phase 2 will end. Therefore, phase 3 will involve a follow-up advertising and promotion campaign to try to keep the regular purchasers from switching to the competitive product. A total of $4 million has been budgeted for this marketing campaign. The problem now is to determine how to allocate this money most effectively to the three phases. Let m denote the initial share of the market (expressed as a percentage) attained in phase 1, f2 the fraction of this market share that is retained in phase 2, and f3 the fraction of the remaining market share that is retained in phase 3. Assume that the money must be spent in integer multiples of $1 million in each phase, where the minimum permissible multiple is 1 for phase 1 and 0 for phases 2 and 3. The following table gives the estimated effect of expenditures in each phase:

3. A company will soon be introducing a new product into a very competitive market and is currently planning its marketing strategy. The decision has been made to introduce the product in three phases. Phase 1 will feature making a special introductory offer of the product to the public at a greatly reduced price to attract first-time buyers. Phase 2 will involve an intensive advertising campaign to persuade these first-time buyers to continue purchasing the product at a regular price. It is known that another company will be introducing a new competitive product at about the time that phase 2 will end. Therefore, phase 3 will involve a follow-up advertising and promotion campaign to try to keep the regular purchasers from switching to the competitive product. A total of $4 million has been budgeted for this marketing campaign. The problem now is to determine how to allocate this money most effectively to the three phases. Let m denote the initial share of the market (expressed as a percentage) attained in phase 1, f2 the fraction of this market share that is retained in phase 2, and f3 the fraction of the remaining market share that is retained in phase 3. Assume that the money must be spent in integer multiples of $1 million in each phase, where the minimum permissible multiple is 1 for phase 1 and 0 for phases 2 and 3. The following table gives the estimated effect of expenditures in each phase:

Practical Management Science
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
Chapter2: Introduction To Spreadsheet Modeling
Section: Chapter Questions
Problem 20P: Julie James is opening a lemonade stand. She believes the fixed cost per week of running the stand...
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3. A company will soon be introducing a new product into a very competitive market and is currently planning its marketing strategy. The decision has been made to introduce the product in three phases. Phase 1 will feature making a special introductory offer of the product to the public at a greatly reduced price to attract first-time buyers. Phase 2 will involve an intensive advertising campaign to persuade these first-time buyers to continue purchasing the product at a regular price. It is known that another company will be introducing a new competitive product at about the time that phase 2 will end. Therefore, phase 3 will involve a follow-up advertising and promotion campaign to try to keep the regular purchasers from switching to the competitive product. A total of $4 million has been budgeted for this marketing campaign. The problem now is to determine how to allocate this money most effectively to the three phases. Let m denote the initial share of the market (expressed as a percentage) attained in phase 1, f₂ the fraction of this market share that is retained in phase 2, and f3 the fraction of the remaining market share that is retained in phase 3. Assume that the money must be spent in integer multiples of $1 million in each phase, where the minimum permissible multiple is 1 for phase 1 and 0 for phases 2 and 3. The following table gives the estimated effect of expenditures in each phase: Stage 3: n = 3 Stage 2 n = 2 Stage 1 n = 1 S3 3 (S3) 0 1 2 3 S2 0 1 2 3 x2 Millions of Dollars Expended Use dynamic programming to determine how to allocate the $4 million to maximize the final share of the market for the new product, i.e., to maximize mf2 f3. Please use the tables given below. X1 S1 4 0 1 2 3 4 1 0 m x3 20 30 40 50 Effect on Market Share 0.2 0.4 0.5 0.6 f₂ (S₂₁x₂) = P₂(x₂) · ƒ3² (S₂ − x₂) 1 2 f3 0.3 0.5 0.6 0.7 - f₁(S₁, x₁) = P₁ (x₁) · ƒ₂² ($₁ − x₁) 3 2 3 4 f2 (S₂) x₂ fi (S₁) x₁
Transcribed Image Text:3. A company will soon be introducing a new product into a very competitive market and is currently planning its marketing strategy. The decision has been made to introduce the product in three phases. Phase 1 will feature making a special introductory offer of the product to the public at a greatly reduced price to attract first-time buyers. Phase 2 will involve an intensive advertising campaign to persuade these first-time buyers to continue purchasing the product at a regular price. It is known that another company will be introducing a new competitive product at about the time that phase 2 will end. Therefore, phase 3 will involve a follow-up advertising and promotion campaign to try to keep the regular purchasers from switching to the competitive product. A total of $4 million has been budgeted for this marketing campaign. The problem now is to determine how to allocate this money most effectively to the three phases. Let m denote the initial share of the market (expressed as a percentage) attained in phase 1, f₂ the fraction of this market share that is retained in phase 2, and f3 the fraction of the remaining market share that is retained in phase 3. Assume that the money must be spent in integer multiples of $1 million in each phase, where the minimum permissible multiple is 1 for phase 1 and 0 for phases 2 and 3. The following table gives the estimated effect of expenditures in each phase: Stage 3: n = 3 Stage 2 n = 2 Stage 1 n = 1 S3 3 (S3) 0 1 2 3 S2 0 1 2 3 x2 Millions of Dollars Expended Use dynamic programming to determine how to allocate the $4 million to maximize the final share of the market for the new product, i.e., to maximize mf2 f3. Please use the tables given below. X1 S1 4 0 1 2 3 4 1 0 m x3 20 30 40 50 Effect on Market Share 0.2 0.4 0.5 0.6 f₂ (S₂₁x₂) = P₂(x₂) · ƒ3² (S₂ − x₂) 1 2 f3 0.3 0.5 0.6 0.7 - f₁(S₁, x₁) = P₁ (x₁) · ƒ₂² ($₁ − x₁) 3 2 3 4 f2 (S₂) x₂ fi (S₁) x₁
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