by creating equal and opposite forces to the resultant x and y forces imbalance forces, expressed through  the following equality constraints /g3533/g1865 /g3036/g1876/g3036/g3041 /g3036/g2880/g2869=/g3398/g1858/g3045 /g2033/g2870cos /g2016 /g3033 , (7)  /g3533/g1865 /g3036/g1877/g3036/g3041 /g3036/g2880/g2869=/g3398/g1858/g3045 /g2033/g2870sin /g2016 /g3033 . (8) where /g1865/g3036is the /g1861/g1872/g1860balancing mass of coordinates /g1876/g3036,/g1877/g3036 /g1853/g1866/g1856 /g1878 /g3036,/g1861 = 1 . . /g1866 ., and  /g2016/g3033is the angle of the  imbalance radial force. Similarly, the moments created by the dynamic imbalance must be counteracted  by the added balancing masses making the combined cross product of inertia nil resulting in the following additional two constraint equations  /g3533/g1865 /g3036/g1876/g3036/g4672/g1878/g3036+/g1859 /g2033/g2870/g4673/g3041 /g3036/g2880/g2869=/g3398/g1839/g3045 /g2033/g2870sin /g2016 /g3014 (9) /g3533/g1865 /g3036/g1877/g3036/g4672/g1878/g3036+/g1859 /g2033/g2870/g4673/g3041 /g3036/g2880/g2869=/g1839/g3045 /g2033/g2870cos /g2016 /g3014 (10) Because the locations of the balancing masses are known, we have a system of four linear equilibrium  Equations (7-10) to /g81unknown mass values  /g1865/g3036 ,/g1861 = 1 … /g1866 . For the case of GMI, we have /g81/g3= 21. There is  no unique solution to this over-determined problem. However, an optimal unique solution may be found  by writing a linear programming problem in which we minimize the sum of the balance masses while meeting the static and dynamic balance equality constraints (7-10) and introducing inequality constraints on the balance masses that take into account the feasibility of actual mass placement with respect to size/mass constraints of the balance mass for a given location and the positive nature of the balance  masses. The objective function to be minimized is expressed as /g2030(/g2195)=/g3533/g2009 /g3036/g1865/g3036/g3041 /g3036/g2880/g2869=/g2235/g2176/g2195 (11)  And the optimization problem is written formally as  min /g3040(/g2235/g2176/g2195)such that /g4676/g1775/g1857/g1869 /g942 /g1813 = /g1802/g1857/g1869 /g2194 /g1854</g2195</g2203 /g1854/g1(12)  The matrix equation /g1775/g1857/g1869 /g942 /g1813 = /g1802/g1857/g1869 is the set of four equality constraints described above and is set up as  shown in Equation (13). /g1743/g1742/g1742/g1742/g1742/g1741/g1876/g2869 /g1876/g2870                    … /g1876 /g3041 /g1877/g2869 /g1877/g2870                    … /g1877 /g3041 /g1876/g2869/g4672/g1878/g2869+/g1859 /g2033/g2870/g4673 /g1877/g2869/g4672/g1878/g2869+/g1859 /g2033/g2870/g4673/g1876/g2870/g4672/g1878/g2870+/g1859 /g2033/g2870/g4673… /g1877/g2870/g4672/g1878/g2870+/g1859 /g2033/g2870/g4673…/g1876/g3041/g4672/g1878/g3041+/g1859 /g2033/g2870/g4673 /g1877/g3041/g4672/g1878/g3041+/g1859 /g2033/g2870/g4673/g1746/g1745/g1745/g1745/g1745/g1744 /g1743/g1742/g1742/g1742/g1742/g1741/g1865/g2869 /g1865/g2870 /g1709 /g1709 /g1709 /g1865/g3041/g1746/g1745/g1745/g1745/g1745/g1744 = /g1743/g1742/g1742/g1742/g1742/g1742/g1742/g1741/g3398/g1858/g3045 /g2033/g2870cos /g2016 /g3033 /g3398/g1858/g3045 /g2033/g2870sin /g2016 /g3033 /g3398/g1839/g3045 /g2033/g2870sin /g2016 /g3014 /g1839/g3045 /g2033/g2870cos /g2016 /g3014/g1746/g1745/g1745/g1745/g1745/g1745/g1745/g1744 (13)  The expression /g2194 /g1854</g2195</g2203 /g1854 Rrepresents a set of 2n inequality constraints that set the lower and upper  bounds on the balance masses. The lower bound must be set to zero, as we are adding mass, and the  308