There are four types of criteria that we will look at.
Zed and Adrian and run a small bicycle shop called "Z to A Bicycles". They must order bicycles for the coming season. Orders for the bicycles must be placed in quantities of twenty (20). The cost per bicycle is $70 if they order 20, $67 if they order 40, $65 if they order 60, and $64 if they order 80. The bicycles will be sold for $100 each. Any bicycles left over at the end of the season can be sold (for certain) at $45 each. If Zed and Adrian run out of bicycles during the season, then they will suffer a loss of "goodwill" among their customers. They estimate this goodwill loss to be $5 per customer who was unable to buy a bicycle. Zed and Adrian estimate that the demand for bicycles this season will be 10, 30, 50, or 70 bicycles with probabilities of 0.2, 0.4, 0.3, and 0.1 respectively.
There are four actions available to Zed and Adrian. They have to decide which of the actions is the best one under each criteria.
Zed and Adrian have control over which action they choose. That is the whole point of decision theory - deciding which action to take.
There are four possible states of nature. A state of nature is an outcome.
Zed and Adrian have no control over which state of nature will occur. They can only plan and make the best decision based on the appropriate decision criteria.
After deciding on each action and state of nature, create a payoff table. The numbers in parentheses for each state of nature represent the probability of that state occurring.
Action | ||||
State of Nature | Buy 20 | Buy 40 | Buy 60 | Buy 80 |
Demand 10 (0.2) |
50 | -330 | -650 | -970 |
Demand 30 (0.4) |
550 | 770 | 450 | 130 |
Demand 50 (0.3) |
450 | 1270 | 1550 | 1230 |
Demand 70 (0.1) |
350 | 1170 | 2050 | 2330 |
Ok, the question on your mind is probably "How the [expletive deleted] did you come up with those numbers?". Let's take a look at a couple of examples.
The opportunistic loss (regret) table is calculated from the payoff table. It is only needed for the minimax criteria, but let's go ahead and calculate it now while we're thinking about it.
The maximum payoffs under each state of nature are shown in bold in the payoff table above. For example, the best that Zed and Adrian could do if the demand was 30 bicycles is to make $770.
Each element in the opportunistic loss table is found taking each state of nature, one at a time, and subtracting each payoff from the largest payoff for that state of nature. In the way we have the table written above, we would subtract each number in the row from the largest number in the row.
Action | ||||
State of Nature | Buy 20 | Buy 40 | Buy 60 | Buy 80 |
Demand 10 | 0 | 380 | 700 | 1020 |
Demand 30 | 220 | 0 | 320 | 640 |
Demand 50 | 1100 | 280 | 0 | 320 |
Demand 70 | 1980 | 1160 | 280 | 0 |
Remember that the numbers in this table are losses and so the smaller the number, the better.
Compute the expected value for each action.
For each action, do the following: Multiply the payoff by the probability of that payoff occurring. Then add those values together. Matrix multiplication works really well for this as it multiplied pairs of numbers together and adds them. If you place the probabilities into a 1x4 matrix and use the 4x4 matrix shown above, then you can multiply the matrices to get a 1x4 matrix with the expected value for each action.
The expected values for buying 20, 40, 60, and 80 bicycles are $400, 740, 720, and 460 respectively. Since the best that you could expect to do is $740, you would buy 40 bicycles.
The maximax criterion is much easier to do than the expected value. You simply look at the best you could do under each action (the largest number in each column). You then take the best (largest) of these.
The largest payoff if you buy 20, 40, 60, and 80 bicycles are $550, 1270, 2050, and 2330 respectively. Since the largest of those is $2330, you would buy 80 bicycles.
The maximin criterion is as easy to do as the maximax. Except instead of taking the largest number under each action, you take the smallest payoff under each action (smallest number in each column). You then take the best (largest of these).
The smallest payoff if you buy 20, 40, 60, and 80 bicycles are $50, -330, -650, and -970 respectively. Since the largest of those is $50, you would buy 20 bicycles.
Be sure to use the opportunistic loss (regret) table for the minimax criterion. You take the largest loss under each action (largest number in each column). You then take the smallest of these (it is loss, afterall).
The largest losses if you buy 20, 40, 60, and 80 bicycles are $1980, 1160, 700, and 1020 respectively. Since the smallest of those is $700, you would buy 60 bicycles.
Here is a table that summarizes each criteria and the best decision.
Action |
|||||
Criterion | Buy 20 | Buy 40 | Buy 60 | Buy 80 | Best Action |
Expected Value | 400 | 740 | 720 | 460 | Buy 40 |
Maximax | 550 | 1270 | 2050 | 2330 | Buy 80 |
Maximin | 50 | -330 | -650 | -970 | Buy 20 |
Minimax | 1980 | 1160 | 700 | 1020 | Buy 60 |
Since there aren't any problems of this kind in the text, work this problem out, and then you can check your answer with the instructor.
Finicky's Jewelers sells watches for $50 each. During the next month, they estimate that they will sell 15, 25, 35, or 45 watches with respective probabilities of 0.35, 0.25, 0.20, and ... (figure it out). They can only buy watches in lots of ten from their dealer. 10, 20, 30, 40, and 50 watches cost $40, 39, 37, 36, and 34 per watch respectively. Every month, Finicky's has a clearance sale and will get rid of any unsold watches for $24 (watches are only in style for a month and so they have to buy the latest model each month). Any customer that comes in during the month to buy a watch, but is unable to, costs Finicky's $6 in lost goodwill.
Find the best action under each of the four decision criteria.