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### Introduction to decision aid

#### Weighting

A decision maker often considers some criteria to be more important than the others. His reasons to think so may be reasonably objective or purely subjective.

A distinction is made between *ordinal* weights, where only the rank is considered (the weights give only a classification of the importance of the criteria), and *cardinal* weights, where the value of each weight against the others represents its relative importance for the decision maker.

These preferences can be expressed in several ways, and need to be formalized properly for each decision aid method. In particular, the values used for the weights information with one method may usually **not** be used directly with another without special care. The reader should therefore understand how the methods work and make sure he supplies the input data (decision matrix and weights) in the proper format before expecting useful results.

Another important aspect to consider is the possible correlation between the criteria: when choosing a car and using the horsepower and fuel consumption as criteria, you are likely to find that the fuel consumption often increases with the horsepower. This means that one of the criteria does not actually help you in your decision process, as it (partially) duplicates the information given by another criterion. This does not mean that such a criterion should be discarded, because it may carry other important information, but it is important to know.

In order to identify the criteria giving similar indications, you can calculate the *criteria correlation matrix*, which contains values between -1 and 1 (0 if there is no correlation, 1 if there is a positive correlation and -1 if there is a negative correlation between criteria).

See DECIDE__CorrelationMatrix() [DECIDE.xls / All functions / CorrelationMatrix].

*Weighted sum*:

The most intuitive way to make a choice between actions, considering several criteria and taking their respective weight in account is to use the *weighted sum* method. With this method, the values used for the weights must be strictly positive, and the weights must be proportional to the preferences (a criterion with a weight of 10 is considered as twice as important as a criterion with a weight of 5).

In order to eliminate the differences caused by the choice of the units for each criterion, the values of the decision matrix and the weights are normalized. Then, for each action, the values obtained for each criterion multiplied by the weight of the criterion, are summed. This gives thus one value by action, and the higher the value, the better the action (see DECIDE__Weight() [DECIDE.xls / All functions / Weight]).

If some of the criteria need to be minimized instead of maximized (like a price, for example), the inverse of the values (1/value) can be used.

*Weighted product*:

The major inconvenient of the *weighted sum* approach is its dependence on the normalization method used. One way to decrease that dependence is to use the *weighted product*. The principle is the same, but all values are multiplied instead of being summed (see DECIDE__Weight() [DECIDE.xls / All functions / Weight]).

Mathematically, this is equivalent to the weighted sum with all the values in the decision matrix replaced by their logarithm. As a direct consequence, the extreme values of the matrix have a very important effect on the results, which is not always acceptable.

*Objective* weighting:

As explained above, the weights can represent the relative importance of the criteria for the decision maker. By definition, they can be extremely subjective, and this is sometimes a problem.

If we consider several criteria, we can analyse the values for each action and calculate how much information each criterion actually supplies: if you compare cars that all cost the same price, the price criterion will not help you much in your decision. Therefore, its weight could be decreased.

Several methods have been proposed to calculate the weights of the criteria depending upon the entropy they show. Basically, the more important the differences between the actions for a criterion, the higher the weight of the criterion.

See DECIDE__ObjectiveWeights() [DECIDE.xls / All functions / ObjectiveWeights].

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