Question

(Value of Information): Midwest Freight Inc. (MFI) is a logistics company handling fragile item shipping. MFI has received an order with the stipulation that if the item arrives late, MFI will pay $1,000 for each late day.

MFI will make $40,000 if the item is delivered on time, but will lose $17,000 if it is defective. MFI can easily deliver the system on time. However, it may, with probability 0.5, be defective. The deficiency can be removed by an adjustment that costs $2,000, but this would delay delivery by 10 days. MFI can also test if the item needs adjustment. However, the test is fallible and costs something. A technical company Bob Ability Inc. (BAI) estimates that there is a 0.6 probability that the test will be favorable (the item does not seem to need adjustment). BAI also thinks that if the test is favorable, there is 25% chance that the item does need adjustment; if the test is unfavorable, this chance increases to 87.5%.

1) Draw the decision tree for MFI.

2) What is the maximum cost of the test that MFI would accept?

Answer 1

1)

2)

I was not clear on the implication of “but will lose $17,000 if it is defective”. Hence I have considered 2 cases, please chose the right interpretation from below. Please do apply the relevant interpretation to payout in abover diagram for defective piece shipped directly without adjustment.

Case 1.Assuming cost to test is T and that all defective pieces are tested. Also assuming that for a defective piece the MFI loses 17000$ out of the 40000$.

Expected payout= (0.25*0.6*0.5+0.875*0.4*0.5)*(28000-T) + (0.75*0.6*0.5+0.125*0.4*0.5)* (40000-T)+ 40000*0.5

=0.25*(28000-T)+ 0.25*(40000-T)+ 20000= 37000- 0.5T

Payout without test= 0.5*(40000-17000)+ 40000*0.5= 11500+20000= 31500

37000- 0.5T>=31500

T<=11000, maximum cost they should pay for test is 11000$

Case 2. Assuming cost to test is T and that all defective pieces are tested. Also assuming that for a defective piece the MFI pays 17000$ additionally to company

Expected payout= (0.25*0.6*0.5+0.875*0.4*0.5)*(28000-T) + (0.75*0.6*0.5+0.125*0.4*0.5)* (40000-T)+ 40000*0.5

=0.25*(28000-T)+ 0.25*(40000-T)+ 20000= 37000- 0.5T

Payout without test= 0.5*(-17000)+ 40000*0.5= 11500

37000- 0.5T>=11500

T<=51000, maximum cost they should pay for test is 51000$