Question

A chain of ski equipment shops purchases skis from a manufacturer each summer for the coming winter season. The most popular intermediate model costs​$175 and sells for ​$300. Any skis left over at the end of the winter are sold at the​ store's spring sale​ (for ​$100​). Sales over the years have been quite stable. Gathering data from all its​ stores, the chain developed the given probability distribution for demand. The manufacturer will take orders only for multiples of 20​, so the chain is considering the following order​ sizes:160​,180​,200​,220​,240.

Construct a payoff table for the ski equipment​ chain's decision problem of how many pairs of skis to order. What is the best decision from an expected value​ basis?

Order $150 175 200 225 250 expected value of profit
160
180
200
220
240
Probability 0.15 0.34 0.29 0.16 0.06
The best decision from an expected value basis is to purchase_____ pairs of​ skis, which gives an expected profit of

​$___. ​



Part 3b. Find the expected value of perfect information.

The expected value of perfect information is ​$____​



Part 4c. What is the expected​ demand?

What is the expected profit if the shop orders the expected​ demand? How does this compare with the expected value​ decision?The expected demand is_____pairs of​ skis, which gives an expected profit of ​$.______This result is____than the expected value decision.

Answer 1

Let’s denote the order sizes as O={160,180,200,220,240} and the demand sizes as D={150,175,200,225,250}. The probability of each demand size is given as P(D)={0.15,0.34,0.29,0.16,0.06}.

The profit for each pair of skis sold is $300 - 175 = $125. Any unsold skis are sold for $100, which results in a loss of $75 per pair of skis. If the demand exceeds the order size, all skis are sold at the regular price.

The payoff for each decision (order size) can be calculated as follows:

If D≤O, the payoff is 125×D−75×(O−D).

If D>O, the payoff is 125×O.

The expected value of each decision is the sum of the payoffs for each demand size, weighted by the probability of that demand size.

The best decision is the order size with the highest expected value.

Part 3b:

The expected value of perfect information (EVPI) is the difference between the expected value with perfect information (the best outcome for each state of nature) and the expected value of the best decision without perfect information.

Part 4c:

The expected demand is the sum of the demand sizes, each weighted by its probability. The expected profit if the shop orders the expected demand can be calculated in the same way as the expected value of each decision.

The result can be compared with the expected value decision to see if ordering the expected demand yields a higher expected profit.

Step-by-step explanation:

This problem involves decision making under uncertainty, where the outcomes of decisions are not known in advance, but probabilities can be assigned to each outcome. The expected value of a decision is a measure of the “average” payoff, weighted by the probabilities of each outcome. The expected value of perfect information measures the maximum amount that a decision maker would be willing to pay for additional information that removes uncertainty. The expected demand is a measure of the “average” demand, weighted by the probabilities of each demand size.