Figure 1

Results of (3) with various values of ρ. From the upper left, the solution is for \((\upsilon _{1},\upsilon _{2},\upsilon _{3},\upsilon _{4})=(2,1,0.4,0.1)\), while the lower left is the optimal outcome for \((1,1,0.9,0.8) \). The upper and lower right graphs represent the solutions for \((1,1,1,0.8) \) and \((1,0.9,1,0.8) \) respectively. We indicate that the cyclic result for these values is based on the fact that \(\mathbb{Q}\) is cyclic