EXPECTED MONETARY VALUES ELEMENTS OF A DECISION ANALYSIS QMBU301 FALL 2012 DECISION MAKING UNDER UNCERTAINTY

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1 QMBU301 FALL 2012 DECISION MAKING UNDER UNCERTAINTY ELEMENTS OF A DECISION ANALYSIS Although there is a wide variety of contexts in decision making, all decision making problems have three elements: the set of decisions (or strategies) available to the decision maker the set of possible outcomes and the probabilities of these outcome a value model that prescribes results, usually monetary values, for the various combinations of decisions and outcomes. Once these elements are known, the decision maker can find an optimal decision. EXPECTED MONETARY VALUES The decision must be made before this uncertainty is resolved. A common way used to make the choice is to calculate the expected monetary value (EMV) of each alternative and then choose the alternative with the largest EMV. EMV is a weighted average of the possible monetary values, weighted by their probabilities. EMV = v i pi 1

2 Decision Trees All of this can be done efficiently using a graphical tool called a decision tree. 1. Decision trees are composed of nodes (circles, squares and triangles) and branches (lines). 2. The nodes represent points in time. A decision node(a square) is a time when the decision maker makes a decision. A probability node (a circle) is a time when the result of an uncertain event becomes known. An end node (a triangle) indicates that the problem is completed - all decisions have been made, all uncertainty have been resolved and all payoffs have been incurred. 3. Time proceeds from left to right. This means that branches leading into a node (from the left) have already occurred. Any branches leading out of a node (to the right) have not yet occurred. TIME 2

3 4. Branches leading out of a decision node represent the possible decisions; the decision maker can choose the preferred branch. Branches leading out of probability nodes represent the possible outcomes of uncertain events; the decision maker has no control over which of these will occur. A a B C b 5. Probabilities are listed on probability branches. These probabilities are conditional on the events that have already been observed (those to the left). Also, the probabilities on branches leading out of any particular probability node must sum to 1. a b given history c DECISION TREE CONVENTIONS This decision tree illustrates these conventions for a single-stage decision problem, the simplest type of decision problem. In a single-stage decision problem all decisions are made first, and then all uncertainty is resolved. Later we will see multistage decision problems, where decisions and outcomes alternate. t Once a decision tree has been drawn and labeled with the probabilities and monetary values, it can be solved easily. 3

4 Folding Back a Decision Tree The solution procedure used to develop this result is called folding back on the tree. Starting at the right on the tree and working back to the left, the procedure consists of two types of calculations. At each probability node we calculate EMV and write it below the name of the node. At each decision node we find the maximum of the EMVs and write it below the node name. After folding back is completed we have calculated EMVs for all nodes. AN EXAMPLE Investing in the stock market An investor has some funds available to invest in one of three choices: a high-risk stock, a low-risk stock, or a savings account that pays a sure $500. If he invests in the stocks, he must pay a brokerage fee of $200. His payoff for the two stocks depends on what happens to the market: If the market goes up, he will earn $1700 from the high-risk stock and $1200 from the low-risk stock. If the market stays at the same level, his payoffs for the high- and low-risk stocks will be $300 and $400, respectively. Finally, if the stock market goes down, he will lose $800 with the high-risk stock but still gain $100 with the low risk stock. FARBUCKS COFFEE MUGS The national coffee store Farbucks needs to decide in August how many holiday-edition insulated coffee mugs to order. These premium mugs sell for $10 and cost $6 each. Farbucks is uncertain of the demand. They believe that there is a 25% chance that they will sell 10,000 mugs, a 50% chance thatt they will sell 15,000 and a 25% chance that they will sell 20,000. Because the mugs are dated, those that are unsold by January 15 are discounted and sold at 20% of the original price. Build a decision tree to determine if they should order 16,000 or 18,000 mugs. 4

5 In June 2010, the operations manager of a large supplier of parts to the automobile industry has been asked if he would take a contract to supply some subassemblies for a demonstration model. The quantity to be ordered was not certain; however it would be either 20 or 40 and the number would be known in January 2011, 7 months from now. The price was $10,000 per unit. He had to respond by next week and commit his company to delivery of the required number of subassemblies by March The operations manager has identified three methods of producing the subassemblies; Process 1 would be the cheapest, if it worked properly. Process 2 was more expensive but sure to work. If Process 1 were used, they would know by September 2010 if it worked. If not there would still be time to use Process 2; however the investment in Process 1 would be lost. With either process, the lead time and other work in the company would require that they commit themselves to a fixed production quantity prior to January. A third possibility was to subcontract the work. A reliable subcontractor was available and if the order were placed now the subcontractor would give them a good price and also be able to wait until the production quantity decision was resolved. If the orders are placed after August 2010, a higher price would be charged but as long as the subcontractor had a firm order on the number of units required by January 2011, they would meet the delivery date. The probability of success with Process 1 was assessed by the engineers who would be involved in the work as 0.5. The operations manager assessed the probability of an order quantity of 40 to be 0.4, after talking with the automobile company. 5

6 Process 1 Testing process $20,000 Incremental costs Per unit production cost if successful 4,000 Process 2 Per unit production cost 6,000 Subcontracting costs (per unit) Order placed before August 1, ,000 Order placed after August 1, ,000 Assumptions: If he produces 20 units and 40 units are ordered, the remaining orders are obtained by subcontracting at $9,000 per unit If he produces 40 units and 20 units are ordered, the excess can be disposed of at $2,000 per unit 6