## 27.5. 27.5 Applications of Pareto games

Let denote again the number of decision makers but suppose now that the decision makers have more than one objective functions separately. There are several possibility to handle such problems:

(A) In the application of multi-objective programming, let denote the weight of the th decision maker, and let be the weights of this decision maker's objective functions. Here denote the number of the th decision maker's objective functions. Thus we can get an optimization problem with the objective function, where all of the decision makers' all the objective functions mean the objective function of the problem, and the weights of the certain objective functions are the sequences. We can use any of the methods from Chapter 27.1 to solve this problem.

(B) We can get another family of methods in the following way. Determine an utility function for every decision maker (as described in Chapter 27.1.1), which compresses the decision maker's preferences into one function. In the application of this method every decision maker has only one (new) objective function, so any methods and solution concepts can be used from the previous chapters.

(C) A third method can be given, if we determine only the partial order of the certain decision makers defined on an alternative set by some method instead of the construction of utility functions. After that we can use any method of Chapter 27.4 directly.

Example 27.18 Modify the previous chapter as follows. Let's suppose again that we choose from four alternatives, but assume now that the committee consists of three people and every member of it has two objective functions. The first objective function is the technical standards of the proposed solution on a subjective scale, while the second one are the odds of the exact implementation. The latter one is judged subjectively by the decision makers individually by the preceding works of the supplier. The data is shown in Figure 27.16, where we assume that the first objective function is judged on a subjective scale from 0 to 100, so the normalized objective function values are given dividing by 100. Using the weighting method we get the following aggregate utility function values for the separate decision makers:

1. Decision maker 2. Decision maker 3. Decision maker Figure 27.16.  The database of Example 27.18. The preferences thus are the following: For example, in the application of Borda measure are given, so the group-order of the four alternatives  Exercises

27.5-1 Let's consider the following table:

Figure 27.17. Let's consider that the objective functions are already normalized. Use method (A) to solve the exercise.

27.5-2 Use method (B) for the previous exercise, where the certain decision makers' utility functions are given by the weighting method, and the group decision making is given by the Borda measure.

27.5-3 Solve Exercise 27.5-2 using the method of paired comparison instead of Borda measure.