For problem 15-26, suppose that if the designer starts working on long designs, she can change them to medium designs after the prevailing style for the season becomes known-although she must then pay a price for this change because of delays in delivery to manufacturers. In particular, if the designer chooses long and the prevailing style is medium, and she then chooses to change to medium, there is a 0.30 chance that her profits will be $200,000 and a 0.70 chance that her profits will be $600,000. No other change from one style to another is possible. Incorporate this information in your decision tree of problem 15-26, and solve the new tree. Give a complete solution in the form of a pair of decisions under given circumstances that maximize the designer's expected profits.
Problem 15-26
Predicting the styles that will prevail in a coming year is one of the most important and difficult problems in the fashion industry. A fashion designer must work on designs for the coming fall long before he or she can find out for certain what styles are going to be "in." A well-known designer believes that there is a 0.20 chance that short dresses and skirts will be popular in the coming fall; a 0.35 chance that popular styles will be of medium length; and a 0.45 chance that long dresses and skirts will dominate fall fashions. The designer must now choose the styles on which to concentrate. If she chooses one style and another turns out to be more popular, profits will be lower than if the new style were guessed correctly. The following table shows what the designer believes she would make, in hundreds of thousands of dollars, for any given combination of her choice of style and the one that prevails in the new season.
Construct a decision tree, and determine what style the designer should choose to maximize her expected profits.