Answer to Question #329501 in Operations Research for Kitta

Question

     Two different products P1 and P2 can be manufactured by one or both of two different machines M1 and M2. The unit processing time of either product on either machine is the same. The daily capacity of machine M1 is 200 units (of either P1 or P2 or a mixture of both) and the daily capacity of machine M2 is 250 units. The shop supervisor wants to balance the products schedule of the two machines such that the total number of units produced on one machine is within 5 units of the number produced on the other. The profit per unit of P1 is $10, and that of P1 is $15. Set up the problem as an LP in equation form.

Answer 1

$x_1,x_2-production\,\,of\,\,P_1\,\,and\,\,P_2\,\,on\,\,M_1\\y_1,y_2\,\,-\,\,production\,\,of\,\,P_1\,\,and\,\,P_2\,\,on\,\,M_2\\\left\{ \begin{array}{c} x_1+x_2\leqslant 200\\ y_1+y_2\leqslant 250\\ \left| x_1+x_2-\left( y_1+y_2 \right) \right|\leqslant 5\\ x_1,x_2,y_1,y_2\geqslant 0\\ 10x_1+10y_1+15x_2+15y_2\rightarrow \max\\\end{array} \right. \\\left\{ \begin{array}{c} x_1+x_2+x_3=200\\ y_1+y_2+y_3=250\\ x_1+x_2-y_1-y_2+z_1=5\\ -x_1-x_2+y_1+y_2+z_2=5\\ x_1,x_2,y_1,y_2,x_3,y_3,z_1,z_2\geqslant 0\\ 10x_1+10y_1+15x_2+15y_2\rightarrow \max\\\end{array} \right.$