Decision trees may be described as the graphic display of the decision-making process. Let us take for example a situation where one must decide whether to go to a movie house or to stay at home and watch TV or a video tape.
State of nature node
New program or cassette
Rerun Good program
The square node signifies a decision point; each alternative is followed by a circular node from which branches on the tree represent the possible outcomes or states of nature which could result.
Strictly speaking, a decision tree must contain both probabilities of outcomes and conditional monetary values of those outcomes so that expected values can be computed.
Let us apply the decision tree in an investment problem. Suppose we are to decide whether to invest our $1000 in stocks or to deposit it in our savings account. Let us further assume that our savings account will not be affected by the performance of the stock market and that it pays an annual interest of 5%. If the market rises, the value of our investment in stock will become $1,400; if the market falls, our investment’s value will decrease to $800. There is a 0.7 probability that the market will rise and a 0.3 probability that the market will fall.
Our decision tree analysis follows:
Market rises 0.7 x $1,400
Market falls 0.3 x $800
Market rises 0.7 x $1,050
Market falls 0.3 x $1,050
The procedure in analyzing a decision tree is to work backward through the tree (from right to left), computing the expected value of each state of nature node. We then choose the particular branch leaving the decision node which leads to the state of nature node with the highest expected value. This process is called rollback or foldback.
Expected value of node 1:
0.7 ( 1,400 ) + 0.3 ( 800 ) = $ 1,220 Expected value of node 2:
0.7 ( 1,050 ) + 0.3 ( 1,050 ) = $ 1,050
Since node 1 has the higher expected value, we must invest our $ 1,000 to shares of stock.
Let us solve a more complicated problem using a decision tree.
Cecilia Industries, Ltd. must decide to build a large or a small plant to produce a new cassette deck which is expected to have a market life of 10 years. A large plant will cost $2,800,000 to build and put into operation, while a small plant will cost only $1,400,000 to build and to put into operation. The company’s best estimate of a discrete distribution of sales over the 10-year period is
Probability = 0.5
Probability = 0.3
Probability = 0.2
Cost-volume-profit analysis done by the management at Cecilia Industries, Ltd. indicates these conditional profits (losses) under the various combinations of plant size and market size:
The opportunity cost of inability to satisfy customer demand is considered in computing the conditional profits.
The decision tree follows
High demand 0.5 x $1,000,000 x 10 yr
Moderate demand 0.3 x $600,000 x 10 yr
Low demand 0.2 x ($200,000) x 10 yr
High demand 0.5 x $250,000 x 10yr
Moderate demand 0.3 x $450,000 x 10yr
Low demand 0.2 x $550,000 x 10yr
Expected value of node 1
(0.5 x $1,000,000 x 10 yr) + (0.3 x $600,000 x 10 yr) + (0.2 x ($200,000) x 10 yr) - $2,800,000*
Expected value of node 2
(0.5 x $250,000 x 10 yr) + (0.3 x $450,000 x 10 yr) + (0.2 x $550,000 x 10 yr) - $1,400,000*
*The costs of building the plants are deducted...
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