Test Bank For Introduction To Operations Research By Frederick Hillier

Test Bank for Chapter 11

Problem 11-1:

A company is planning its advertising strategy for next year for its three major products. Since the three products are quite different, each advertising effort will focus on a single product. In units of millions of dollars, a total of 6 is available for advertising next year, where the advertising expenditure for each product must be an integer greater than or equal to 1. The vice-president for marketing has established the objective: Determine how much to spend on each product in order to maximize total sales. The following table gives the estimated increase in sales (in appropriate units) for the different advertising expenditures.

Advertising Expenditure

Product

1 2 3

1

7 4 6

2

10 8 9

3

14 11 13

4

17 14 15

Use dynamic programming to solve this problem.

Solution for Problem 11-1:

Since the decisions to be made are the advertising expenditures on the three products, the stages for a dynamic programming formulation of this problem correspond to the three products. When making the decision for a particular product, the essential information is the amount of the advertising budget still remaining, so this becomes the current state in this formulation.

Let x_n be the advertising dollars (in millions) spent on product n.

Let s_n be the amount of advertising budget remaining.

Let be the increase in sales of product n when x_n million dollars are spent on product n, as given by the above table.

Then, using the usual dynamic programming notation presented in Sec. 10.2 of the textbook, the recursive relationship for this problem is

.

The solution procedure now starts at the end (stage 3) and moves backward stage by stage.