Deborah Hollwager a concessionaire for the Des Moines ballpa

Deborah Hollwager, a concessionaire for the Des Moines ballpark, is trying to decide how much hotdogs to order for an upcoming Rock Fest event. She has developed a table of conditional values for the various alternatives (stocking decision) with payoffs, and states of nature (size of crowd): Suppose Deborah knows nothing about the likelihood of states of nature, how can she make decisions about stock level? Show details of your reasoning and analysis. If Deborah knows that the probabilities associated with the states of nature are 0.3 for larege crowd, 0.5 for medium crowd, and 0.2 for a small crowd, determine the best stock level. What is the max EMV? Use the decison tree approach to solve b). Suppose now Deborah has the opportunity of getting perfection information about states of nature, but it costs her $1,200. Will it be worthwhile for Deborah to get such perfect information? Why? Show details of your reasoning and solution steps.

Solution

This is Decision analysis problem about making a choice about three alternatives: large stock, medium stock and small stock. Similarly there are three nature of states that may emerge during Rock festival as large crowd, medium crowd and small crowd

Pay-off matrix along with Maximum, Minimum and Average gains for each options are given in the following table

a) In the case of having no knowledge about likelyhood of the nature of the states, Dehborah may follow optimistic approach and make choice on the basis of MaxiMax that is go far Large stocks or pessimistic approach of Maximin or Minimin and respectively go for Medium and Small socks. Lastly assuming all outcomes equally likely decide about the best option as Medium Stocks that gives the maximum average.

b) EMV (expected monitary value) is calculated by using the expected value formula by using the probabilities given for each state under each options as follows:

EMV for Large Stocks = .3*20,000 +.5*10,000 + .2*(-2,000) = 10,600

EMV for Medium Stock = .3*15000 + .5*12,000 +.2*6000 = 11,700

EMV for Small Stock = .3*9000 + .5*6,000 +.2*5,000 = 6,700

Therefore the Maximum EMV is $11,700 in favour of Medium Stock

c) Decision tree has one decision node with three options (alternate branches) as Large Stock, Medium Stock and Small Sock and followed by state nodes having three branches as large crowd, medium crowd and small crowd having payoffs and probabilities as mentioned above. Analysis is as per the calculations shown in b) above.

d) Expected value under perfect information = .3*20,000 + .5*12000 + .2*6000 = 13,200 ( sum of product of probabity of each state with the maximum value under the state)

Therefore maximum cost for perfect information = Expected EMV under perfect information - Maximum EMV without perfect information = 13,200 - 11,700 = 1,500

Therefore it is worthwhile to have the perfect information at cost of $1,200

Alternatives	Large crowd	Medium Crowd	Small Crowd	Maximum	Minimum	Average
Large Stock	$20,000	$10,000	-$2000	$20000	-$2,000	$9333.33
Medium Stock	$15,000	$12,000	$6,000	$15,000	$6,000	$11,000
Small Stock	$9,000	$6,000	$5,000	$9,000	$5,000	$666.67