Determining compromise weights for group decision making

In our early work, we described a minimax principle based procedure of preference adjustments with a finite number of steps to find compromise weights. The weights are obtained by solving a linear programming (LP) problem. The objective value of the LP is called a compromise index. When the index is non-negative, compromise weights are determined; otherwise we identify a set of 'worst preference orders' according to the optimal solution of the LP, for adjustment. However, this 'worst preference order' set depends on the selection of corresponding optimal solutions. This may have a negative impact on the decision-making procedure. This paper thoroughly discusses the problem of the existence of multiple optimal solutions. We define a set of 'very worst preference order', which is independent of the selection of optimal solutions. We prove that compromise weights can be achieved within a finite number of adjustments on preference orders. Numerical examples are given for illustration.