In the previous chapter we assumed that the objective functions are given by numerical values. These numerical values also mean preferences, since the th decision maker prefers alternative to , if . In this chapter we will discuss such methods which don't require the knowledge of the objective functions, but the preferences of the certain decision makers.
Let denote again the number of decision makers, and the set of decision alternatives. If the th decision maker prefers alternative to , this is denoted by , if prefers alternative to or thinks to be equal, it is denoted by . Assume that
(i) For all , or (or both)
(ii) For and , .
Condition (i) requires that the partial order be a total order, while condition (ii) requires to be transitive.
We illustrate the definition of group decision-making function by some simple example.
Let given positive constant, and
The group decision-making function means:
The majority rule is a special case of it when .
This kind of group decision-making is also called dictatorship.
Example 27.14 In the case of Borda measure we assume that is a finite set and the preferences of the decision makers is expressed by a measure for all . For example , if is the best, , if is the second best alternative for the th decision maker, and so on, , if is the worst alternative. Then
A group decision-making function is called Pareto or Pareto function, if for all and , necessarily. That is, if all the decision makers prefer to , it must be the same way in the collective preference of the group. A group decision-making function is said to satisfy the condition of pairwise independence, if any two and preference structure satisfy the followings. Let such that for arbitrary , if and only if , and if and only if . Then if and only if , and if and only if in the collective preference of the group.
Example 27.15 It is easy to see that the Borda measure is Pareto, but it doesn't satisfy the condition of pairwise independence. The first statement is evident, while the second one can be illustrated by a simple example. Be , . Let's assume that
Then , thus . However , so . As we can see the certain decision makers preference order between and is the same in both case, but the collective preference of the group is different.
Let denote the set of the -element full and transitive partial orders on an at least three-element set, and be the collective preference of the group which is Pareto and satisfies the condition of pairwise independence. Then is necessarily dictatorial. This result originated with Arrow shows that there is no such group decision-making function which could satisfy these two basic and natural requirements.
Example 27.16 The method of paired comparison is as follows. Be arbitrary, and let's denote the number of decision makers, to which . After that, the collective preference of the group is the following:
that is if and only if more than one decision makers prefer the alternative to . Let's assume again that consists of three elements, and the individual preferences for
Thus, in the collective preference , because and . Similarly , because and , and , since and . Therefore which is inconsistent with the requirements of transitivity.
The methods discussed so far didn't take account of the important circumstance that the decision makers aren't necessarily in the same position, that is they can have different importance. This importance can be characterized by weights. In this generalized case we have to modify the group decision-making methods as required. Let's assume that is finite set, denote the number of alternatives. We denote the preferences of the decision makers by the numbers ranging from 1 to , where 1 is assigned to the most favorable, while is assigned to most unfavorable alternative. It's imaginable that the two alternatives are equally important, then we use fractions. For example, if we can't distinguish between the priority of the 2nd and 3rd alternatives, then we assign 2.5 to each of them. Usually the average value of the indistinguishable alternatives is assigned to each of them. In this way, the problem of the group decision can be given by a table which rows correspond to the decision makers and columns correspond to the decision alternatives. Every row of the table is a permutation of the numbers, at most some element of it is replaced by some average value if they are equally-preferred. Figure 27.11 shows the given table in which the last column contains the weights of the decision makers.
In this general case the majority rule can be defined as follows. For all of the alternatives determine first the aggregate weight of the decision makers to which the alternative is the best possibility, then select that alternative for the best collective one for which this sum is the biggest. If our goal is not only to select the best, but to rank all of the alternatives, then we have to choose descending order in this sum to rank the alternatives, where the biggest sum selects the best, and the smallest sum selects the worst alternative. Mathematically, be
for . The th alternative is considered the best by the group, if
The formal algorithm is as follows:
Applying the Borda measure, let
and alternative is the result of the group decision if
The Borda measure can be described by the following algorithm:
Applying the method of paired comparison, let with any
which gives the weight of the decision makers who prefer the alternative to . In the collective decision
In many cases the collective partial order given this way doesn't result in a clearly best alternative. In such cases further analysis (for example using some other method) need on the
non-dominated alternative set.
By this algorithm we construct a matrix consists of the elements, where if and only if the alternative is better in all then alternative . In the case of draw .
Example 27.17 Four proposal were received by the Environmental Authority for the cleaning of a chemically contaminated site. A committee consists of 6 people has to choose the best proposal and thereafter the authority can conclude the contract for realizing the proposal. Figure 27.12 shows the relative weight of the committee members and the personal preferences.
Using the majority rule
so the first alternative is the best.
Using the Borda measure
In this case the first alternative is the best as well, but this method shows equally good the second and third alternatives. Notice, that in the case of the previous method the second alternative was better than the third one.
In the case of the method of paired comparison
Thus and . These references are showed by Figure 27.13. The first alternative is better than any others, so this is the obvious choice.
In the above example all three methods gave the same result. However, in several practical cases one can get different results and the decision makers have to choose on the basis of other criteria.
Apply the majority rule.
27.4-3 Apply the method of paired comparison to Exercise 27.4-1.
Repeat Exercise 27.4-1 for this exercise.
27.4-6 Apply the method of paired comparison to Exercise 27.4-4.