1. IE 342 Decision Tree Examples

2. Example 1
You are considering marketing a new product. Once the product is introduced, there is a 70% chance of encountering a competitive product. Two options are available in each situation:
Option 1 (with competitive product): In this situation, you have two options; you can raise or lower your price and see how your competitor responds.
If you raise your price, your competitor can raise or lower its price with equal probability. If the competitor raises its price, your profit will be $60, and if the competitor lowers its price, you will lose $20.
If you lower your price, your competitor can again raise or lower its price. With a probability of 20% it raises its price and your profit will be $40 and with a probability of 80%, it lowers its price and your profit will be $10.
Option 2 (without competitive product): You still have two options. If you raise, your profit will be $100 and if you lower your profit will be $30.
Construct decision tree, determine payoffs and find out decision rules.

3. Example 2
A company is considering purchasing of a new labor-saving machine. The machine’s cost will turn out to be $55/day. Each hour of labor that is saved reduces cost by $5. However, there is some uncertainty over the number of hours that actually will be saved.
It is judged that the hours of labor saved per day will be 10 hrs, 11 hrs or 13 hrs, with probabilities of 0.1, 0.6 and 0.3, respectively.
Construct a decision tree, determine payoffs and find out decision rules to maximize the profit.
Define “profit” as the excess of labor-cost savings over the machine cost (i.e. profit = labor cost savings-machine costs).

4. Example 3 (Bill’s Investment Problem)
Suppose Bill has $50,000 to invest in the financial market for one year. His choices have been narrowed to two options:
Option 1 (Stock): Buy 1,000 shares of a stock at $50/share that will be held for one year. There is a brokerage fee of $100 for the transaction (for either buying or selling stocks). For simplicity, assume that, the stock is expected to provide a return at any one of three different levels:
a high level (A) return of $25,000 with probability 0.25.
a medium level (B) return of $4,500 with probability 0.4.
a low level(C) return (loss) of -$15,000 with probability 0.35.
No stock dividend is anticipated for this stock.
Option 2 (Bond): Purchase a $50,000 bond, which has a return of $3,750. The interest earned from this bond is nontaxable income. However, there is a $150 transaction fee for either buying or selling the bond.
Which alternative should be selected to maximize Bill’s gain. Assume that any long-term capital gains will be taxed at 20%. Bill’s MARR is known to be 5% after taxes. Determine the payoff amount at the tip of each branch.