1. A local daily newspaper has recently been acquired by a large media conglom- erate. The paper currently sells for $1.50/week and has a circulation of 80,000 subscribers. Advertising sells for $250/page, and the paper currently sells 350 pages/week (50 pages/day). The new management is looking for ways to increase profits. It is estimated that an increase of ten cents/week in the subscription price will cause a drop in circulation of 3,000 subscribers. Increasing the price of ad- vertising by $100/page will cause the paper to lose approximately 60 pages of advertising per week. The loss of advertising will also affect circulation, since one of the reasons people buy the paper is for the advertisements. It is estimated that a loss of 60 pages of advertisements per week will reduce circulation by 1,000 subscriptions. Management has decided not to increase the price of advertising beyond $395/page and not to increase the price of the paper beyond $2.60/week. (a) Formulate the problem as a constrained non-linear optimization model and find the weekly subscription price and the advertising price that will maxi- mize profit. What is the maximum profit? (b) Examine the sensitivity of your conclusions in part (a) to the assumption of 3,000 lost sales when the price of the paper increases by ten cents. (c) Examine the sensitivity of your conclusions in part (a) to the assumption of 60 pages/week of lost advertising sales when the price of advertising is increased by $100/page. (d) What is the value of the multiplier (Lagrange multiplier) corresponding to the optimal solution found in part (a)? What is the real world significance of the multiplier?
1. A local daily newspaper has recently been acquired by a large media conglom- erate. The paper currently sells for $1.50/week and has a circulation of 80,000 subscribers. Advertising sells for $250/page, and the paper currently sells 350 pages/week (50 pages/day). The new management is looking for ways to increase profits. It is estimated that an increase of ten cents/week in the subscription price will cause a drop in circulation of 3,000 subscribers. Increasing the price of ad- vertising by $100/page will cause the paper to lose approximately 60 pages of advertising per week. The loss of advertising will also affect circulation, since one of the reasons people buy the paper is for the advertisements. It is estimated that a loss of 60 pages of advertisements per week will reduce circulation by 1,000 subscriptions. Management has decided not to increase the price of advertising beyond $395/page and not to increase the price of the paper beyond $2.60/week. (a) Formulate the problem as a constrained non-linear optimization model and find the weekly subscription price and the advertising price that will maxi- mize profit. What is the maximum profit? (b) Examine the sensitivity of your conclusions in part (a) to the assumption of 3,000 lost sales when the price of the paper increases by ten cents. (c) Examine the sensitivity of your conclusions in part (a) to the assumption of 60 pages/week of lost advertising sales when the price of advertising is increased by $100/page. (d) What is the value of the multiplier (Lagrange multiplier) corresponding to the optimal solution found in part (a)? What is the real world significance of the multiplier?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please, help solve this modeling problem ! Constrained Non-linear optimization model !
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1. A local daily newspaper has recently been acquired by a large media conglom-
erate. The paper currently sells for $1.50/week and has a circulation of 80, 000
subscribers. Advertising sells for $250/page, and the paper currently sells 350
pages/week (50 pages/day). The new management is looking for ways to increase
profits. It is estimated that an increase of ten cents/week in the subscription price
drop in circulation of 3, 000 subscribers. Increasing the price of ad-
vertising by $100/page will cause the paper to lose approximately 60 pages of
advertising per week. The loss of advertising will also affect circulation, since
one of the reasons people buy the paper is for the advertisements. It is estimated
that a loss of 60 pages of advertisements per week will reduce circulation by 1, 000
subscriptions. Management has decided not to increase the price of advertising
beyond $395/page and not to increase the price of the paper beyond $2.60/week.
will cause a
(a) Formulate the problem as a constrained non-linear optimization model
and find the weekly subscription price and
mize profit. What is the maximum profit?
advertising price that will maxi-
(b) Examine the sensitivity of your conclusions in part (a) to the assumption
of 3,000 lost sales when the price of the paper increases by ten cents.
(c) Examine the sensitivity of your conclusions in part (a) to the assumption
of 60 pages/week of lost advertising sales when the price of advertising is increased
by $100/page.
(d) What is the value of the multiplier (Lagrange multiplier) corresponding
to the optimal solution found in part (a)? What is the real world significance of
the multiplier?
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