Decision trees : applications• When the user has an objective he is trying to achieve: max. profit, optimise cost• When there are several courses of action• There is a calculable measure of benefit of the various alternatives• When there are events beyond the control of the decision maker: environmental factors• Uncertainty concerning which outcome will actually happen
• A fruit vendor sells strawberries. One case of strawberries costs him $20 which he sells for $50 per case. If not sold on the same day, it is worthless. Analysis of past data shows that the demand for cases is as per the table attached.• What is the optimal stock he should order?
Daily sales No of days sold Probability10 15 0.1511 20 0.2012 45 0.4513 25 0.25Total 100 1.00
Expected profit• Expected profit if he stocks 10 cases is:300x0.15+ 300x0.20+300x0.40+300x0.25=300If he stocks 11 cases: exp profit=322.50 12 cases: exp profit =335 13 cases: exp profit =327.50Optimal course of action is to stock 12 cases
• If he had perfect information, then he would stock the exact number of cases required each day. The actual demand still varies, but he knows it in advance.• In this case the conditional profit will be the maximum profit for each level of demand.• Expected profit with perfect information will be:• 300x0.15+330x0.20+360x0.40+390x0.25= 352.50• So with perfect information he makes an additional profit of 352.50-335=17.50• This is the maximum price he will be willing to pay for ‘perfect information’
Decision tree analysis• A graphic model of the decision process• Useful in making decisions concerning investments, project management• Squares symbolize decision points• Circles represent chance events
• When analyzing decision trees:i) Start from the right (top of the tree) and work back to the left (root)ii) When analyzing a chance node (circle) calculate the expected value at that node by multiplying the probability on each branch emanating from that node by the profit at the end of that branch and then summing up for all the branches that emanate from that nodeiii) When analyzing a decision node (square), let the expected value at that node be the maximum of the expected values for all the branches emanating from that node.iv) In this way, we choose the action with the largest expected value while pruning the branches corresponding to the less profitable actions
• Christie has recently received an offer from a large hotel chain to operate the resort for the winter , guaranteeing her a $ 45000 for the season. She is also considering leasing snow making equipment for the season.If the equipment is leased, the resort will be able to operate full-time , regardless of the amount of natural snowfall. If she decides to use snowmakers to supplement the natural snowfall, her profit for the season will be$1,20,000 minus the cost of leasing and operating the equipment. The leasing cost will be $ 12,000 per season irrespective of how much it is used. The operating cost will be $10,000 if the snowfall is more than 40 inches, $ 50000 if the snowfall is between 20 and 40 inches and $ 90,000 if it is less than 20 inches.• The probability distribution of the snowfall and the resulting profit is summarized in the attached table• What should Christie do?
Amount of snow Profit ProbabilityMore than 40 1,20,000 0.4inchesBetween 20 and 40 40,000 0.2inchesLess than 20 inches -40,000 0.4