Jovan's Movers rents out trucks with a crew of two on a daily basis, usually to homeowners who are moving or to companies with delivery problems. On one particular day Jovan is a truck short and intends to hire one from a local truck rental firm. However, he does not know how large the load is that needs to be moved. How big a truck should he rent? A large truck costs $200 per day (including insurance, fuel, etc.), a small truck $130 per day. A small truck is cheaper but if the load is too large, the crew may have to make two trips. Jovan assesses the additional cost of making two trips (overtime and truck mileage) at $150 beyond the costs for a single trip. He assesses the probability that two trips will be necessary if he rents a small truck at 0.40. Assume that if Jovan rents a large truck it can accommodate any size load in a single trip. a. Assuming there are no other ramifications to the decision, should Jovan rent a large truck or a small truck? Construct a decision tree (manually or using PrecisionTree) to support your answer and explain your recommendation. Would your answer change if the probability that two trips will be necessary is 50% instead of 40%?b. What is the most Jovan would pay to know for sure whether a small truck or a large truck would be adequate for the job? For example, suppose he could hire someone to inspect the contents of the move in advance. Construct a second decision tree to support your answer and estimate if the probability of needing two trips with a small truck is set to 40% as in part a).c. Suppose Jovan is risk averse, with a risk tolerance value of $1,000 (assume the exponential utility function applies). Would this change your answer to part a)?

Explanation:

In this problem business of Jovan is to rent out trucks and earn revenues. On a particular day there is a shortage of one truck. It can be taken on rent from other party. If a big truck is hired, then any load can be carried. But the rental cost is $200. Small truck cannot carry weight beyond a range. In that case two trips are needed. Rental of one trip of small truck is $130. Cost of two trip is $150 extra. So it is $130+$150=$280. Probability of two trips is 40%. So based on these data, following decision tree diagram is draw:

From this decision tree expected rental cost of small truck based on probability is-

Expected rental of small truck =0.6 x $130 + 0.4 x $280

                                                                =$78+\$112

                                                                 =$190

Decision: Since expected rental of small truck is $190, it is lower than rental of big truck of $200. So small truck is recommended.

If probabilities of trips are 50:50, then expected rental of small truck is-

Expected rental of small truck =0.5 x $130 + 0.5 x $280

                                    =$65 + $140

                                    =$205

Now it is more than rental of big truck. So hiring of big truck is recommended.

b) Now Jovan wants to hire an outside consultant. He will assess and recommend whether to hire a big truck or a small truck. If he recommend for big truck, then big truck will be hired. Otherwise a small truck will be bought. As per current situation probability of two trip is 40%. If consultant approves this situation, then big truck will be hired. Thus probability of hiring big truck is 40% under recommended scenario. So probability of hiring small truck with one trip is 60%. On this basis decision chart is drawn below:

Based on this diagram, expected cost of hiring a truck is-

Expected rental =0.4 x $200 + 0.6 x $130

                          = $80 + $78

                          = $158

If you compare this expected cost with the expected cost of $190 in part (a), then it is lower by $190-$158=$32

Hence, maximum $32 can be paid to consultant for hiring and taking perfect decision.

c) Now Jovan has been taken as risk averser. His risk tolerance value is $1,000. Suppose utility function is exponential of following form-

U=e^{P} where p is the probability of two trips by small truck

As a risk averser he will undertake risk only when this U value is $1,000.

U=e^{P} = $1,000

Take log on both side to get-

Plog e =  log1,000

{P}{log}2.71828 =  log1,000 [ since e =2.71828]

{P}= 3 / 0.43429189

    =6.929 percent

So the risk averse Jovan will go for small truck only when probability of two trips for small car is 6.929 percent. Here it is 40%. So big truck will be hired.