Decision theory: decision matrix
Posted by: carkod on: October 5, 2008
The decision problem
Usually we have to choose an alternative among others, the act of deciding may carry certainty or uncertainty. Decisions are for certain when you´ve got plenty information of what you´ll do, for instance, if we want to buy clothes there´s a lot of shops, cheaper or more expensive, this decision has perfect information, so you understand the main facts that characterizes each shop (prices, quality, brands…) Otherwise, decisions could be unclear, and you´ll have to determine what you´ll choose.
A decision is a process in which you make a decision considering the consequences. These consequences are: the state of nature, which are events uncontrollable by the decider. States of nature are mutually exclusive {raining, not raining} it´s impossible to see both events to happen; and exhaustive {raining, not raining, snowing} this is out of our interest because we´re only considering “raining”. The decision (go swimming) is influenced by the states of nature coming up different kinds of outcomes depending on state of nature (go swimming or not). Another fact is the possibility of occurrence, which depends on your information or your preferences.
Decision matrix
The decision matrix is written with the two variables mentioned before, that is, the decision (di) and the state of nature (ei), where the first one are the columns, and the state of nature are rows (assuming i =1,2,3..).
For example, a decision maker has to choose to do an investment. An investment on low quality computers (d1) could cause low demand (e1) which will give us 40 (in thousands of $) revenues, comparative demand (e2) which will give us losses of 10 or high demand(e3) will provide 0 revenue. An investment on intermediate quality computers (d2) could cause a low demand (e1) giving us 0 revenue, comparative demand (e2) will provide 30 revenue and high demand (e3) 5 revenue. An investment on high quality computers (d3) could cause a low demand (e1) with 100 losses, comparative demand (e2) will give out 10 revenues, and high demand (e3) will provide 20 revenues. With this information the decision matrix A would be as follows:
We could also add to the matrix the percentage of probability of each state of nature, the vector of probabilities (P).
Decision theory: the decision criteria
Posted by: carkod on: October 5, 2008
Decision criteria
Choosing something is just like a lottery, but to make it less random, we should follow some selection criteria in order to choose among the alternatives helpings us to compare and of course make our decision.
The minimum and maximum criteria
This trick is pretty easy, suitable for pessimistic people. Just take the lowest value of each column (di) and compute the maximum of the lowest values you just took. Let´s see the matrix before.
The minimum of each decision column (di) is (first step):
The maximum of the minimum is (second step):
The maximum and maximum criteria
This criterion has an optimistic sight of the decision. So take the biggest value and compute the maximum among the biggest values, as follows (first step):
Secondly, calculate the maximum of these values, which are:
However, these two criteria do not match the interests and preferences (probabilities of carrying out the decision) of the decision maker. They only choose the best decision according to their real values.
The expected value criteria
For the expected value we use the vector P, which are the probabilities of carrying out the decision. Multiply by the probabilities each decision (according to each state of nature) and take out the biggest value (maximum). The highest value represents the best decision, and the value obtained through the computation is the expected earnings from the decision.
In the example, where p(ei) is the probabilities vector, and aij each demand influenced by correspondent state of nature, and “m” the matrix size:
Decision tree
Posted by: carkod on: October 5, 2008
Decision tree
Another method to depict the sequence of a decision is the decision tree. Where squares are decisions and circles are states of nature (events). We usually depict two contrary events, connected by branches. Over the braches probabilities of occurrences are written.
Here, “A” means bet, “G” means win and “P” to lose. The respective braches represents probabilities of losing or wining. To analyze this type of sequence we do backward induction, that is, we solve the expected value in every decision branch and we compare the result.
Decision theory: perfect information
Posted by: carkod on: October 5, 2008
Perfect information
An individual with perfect information knows before the decision that a state of nature is going to happen, the best choice would be the maximum of each state of nature (each row ei), this is usually denoted by Mi. Afterwards, we calculate the expected value, under the name of expected value on perfect information (EVPI).From the previous example, we have:
And in order to find the value of perfect information just subtract expected value of perfect information to expected value of the decision (the ultimate decision).
Perpetual and periodic inventory system
Posted by: carkod on: September 1, 2008
Perpetual and periodic inventory system
Perpetual inventory system records every transaction. Inventory must be reordered once a year. Suitable for all kind of goods. Firm 1 buys from firm 2 Process:
Periodic inventory system records only cash transactions. Inventory must be reordered quarterly. Suitable only for invaluable goods. Process:
In periodic system there´s no cost of goods sold, inventory is only updated for financial statements.
