Consider a GDMP, where three decision-makers try to sort out one from three candidate solutions as scoring in Table 1.

When [f.sup.*.sub.i] = -[infinity], a special scenario, there is no optimal concession equilibrium solution to GDMP. In the following, it is assumed that [f.sup.*.sub.i] > -[infinity], i = 1, 2, ....

Then, for any given [bar.x] [member of] X, [bar.x] is an [bar.s]-concession equilibrium solution to GDMP, where [bar.s] = max{[f.sub.i]([bar.x]) - [f.sup.*.sub.i] - [[epsilon].sub.i] | i = 1, 2, ..., r}.

From Lemma 7, it is known that, for any x [member of] X, there exists an equilibrium value [epsilon] of GDMP at the concession value s, which makes the point x be an s-concession equilibrium solution to GDMP at the concession value [epsilon].

Then [x.sup.*] is [s.sup.*]-optimal concession equilibrium solution to GDMP at the value [epsilon] if and only if ([x.sup.*], [s.sup.*]) is an optimal solution to (S).

So, by Definition 1, we have that [x.sup.*] is an [s.sup.*]-concession equilibrium solution to GDMP at the value [epsilon].

That is, [bar.s] = [s.sup.*] and [x.sup.*] is the [s.sup.*]-optimal concession equilibrium solution to GDMP at the value [epsilon].

Now, assume that [x.sup.*] is the [s.sup.*]-optimal concession equilibrium solution to GDMP at the value [epsilon]; then, by Definition 1, we know that ([x.sup.*], [s.sup.*]) is a feasible solution to (S).

In fact, just assume that X is a compact set and [f.sub.i](i = 1, 2, ..., r) is a continuous function on X; then, according to Lemma 7, it is known that the [s.sup.*]-optimal concession equilibrium solution to GDMP exists.

For k = 1, 2, ..., let [x.sup.*.sub.k] [member of] X bean [s.sub.k]-concession equilibrium solution to GDMP. Because X is compact, the sequence {[x.sup.*.sub.k]} has a convergent subsequence.

According to Theorem 8, we obtain a method to find out an [s.sup.*]-optimal concession equilibrium solution to GDMP. First, the optimal solution to the problem (Pi) is solved, respectively.

If [([x.sup.*], [bar.s]), ([x.sup.1*], [x.sup.2*], ..., [x.sup.r*])] is an optimal solution to ([bar.S]), then [x.sup.*] is an [s.sup.*]-optimal concession equilibrium solution to GDMP at the value [epsilon], where [s.sup.*] = [bar.s] + [??].