Answer to Question #200371 in Operations Research for lHEN

Question

A company that produces two kinds of office tables, T1 and T2. It takes 2 hours to produce the parts of one unit of T1, 1 hour to assemble and 2 hours to polish. It takes 4 hours to produce the parts of one unit of T2, 2.5 hour to assemble and 1.5 hours to polish. Per month, 7000 hours are available for producing the parts, 4000 hours for assembling the parts and 5500 hours for polishing the tables. The profit per unit of T1 is $90 and per unit of T2 is $110. cOMPUTE THE DUAL PRICE.

Answer 1

optimization problem is

primal problem

$maximize \space 90x+110y\\ subject \ to,\\ 2x+4y\le 7000\\ x+2.5y \le 4000\\ 2x+1.5y\le5500$

convert primal problem to dual problem

$Min \space Z_d=7000w_1+4000w_2+5500w_3\\ subject \space to\space 2w_1+w_2+2w_3≥90\\ 4w_1+2.5w_2+1.5w_3≥110 \\ and w_1,w_2,w_3≥0;$

now we solve dual problem with BIG-M method

$Min \space Z = 7000 w_1 + 4000 w_2 + 5500 w_3 + 0 w_1 + 0 w_2 + MA_1 + M A_2\\ subject \space to\\ 2 w_1 + w_2 + 2 w_3 - w_1 + A_1 = 90\\ 4 w_1 + 2.5 w_2 + 1.5 w_3 - w_2 + A_2 = 110\\ and w_1,w_2,w_3,w_1,w_2,A_1,A_2≥0$

$Positive\space max\space Z_j-C_j\space is\space 6M-7000\space and\space its\space column\space index\space is\space 1.\\ \space So,\space the\space entering\space variable\space is\space w_1.\\ Minimum\space ratio\space is\space 27.5\space and\space its\space row\space index\space is\space 2.\space \\ So,\space the\space leaving\space basis\space variable\space is\space A_2. \\∴\space The\space pivot\space element\space is\space 4.\\ Entering\space =w_1,\space Departing\space =A_2,\space Key\space Element\space =4$

$Positive\space max\space Z_j-C_j\space is\space 1.25M-2875\space and\space its\space column\space index\space is\space 3.\\ \space So,\space the\space entering\space variable\space is\space w_3.\\ Minimum\space ratio\space is\space 28\space and\space its\space row\space index\space is\space 1.\space \\ So,\space the\space leaving\space basis\space variable\space is\space A_1. \\∴\space The\space pivot\space element\space is\space 1.25.\\ Entering\space =w_3,\space Departing\space =A_1,\space Key\space Element\space =1.25$

$Since \space all\space Zj-Cj≤0\\ Hence, optimal \space solution \space is\space arrived\space with\space value\space of\space variables\space as :\\ w_1=17,w_2=0,w_3=28\\ Min Z=273000\\ we \space can\space write \space solution \space for \space primal \space problem\\$

maximum profit $273000 by making type T₁ table 2300 and type T₂ table 600