document.write( "Question 1102103: I have no idea how to calculate this. Appreciate the help. Thanks.
\n" ); document.write( "Solve the following linear programming problem.
\n" ); document.write( "Minimize: z=5x+30y
\n" ); document.write( "subject to: 9x+12y≥61
\n" ); document.write( " 8x+4y≥32
\n" ); document.write( " x≥0, y≥0
\n" ); document.write( "What is the minimum value of z? \n" ); document.write( "

Algebra.Com's Answer #716814 by ikleyn(52788) $\"\"$ $\"About$

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document.write( "First inequality 9x + 12y >= 61 is THIS restriction\r\n" );
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document.write( "     y >= .\r\n" );
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document.write( "Second inequality 8x + 4y >= 32 is THIS restriction\r\n" );
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document.write( "    y >= .\r\n" );
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document.write( "Together with the inequalities x >= 0, y >= 0 they form THIS feasibility domain in the first quadrant QI, shown in the Figure below:\r\n" );
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document.write( "Plot y >=  (over the read line) and  y >=  (over the green line)\r\n" );
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document.write( "Feasibility domain is INFINITE AREA in Q1 OVER the both red and green lines.\r\n" );
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document.write( "Feasibility domain has 3 vertices:\r\n" );
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document.write( "    P1 = (0,8)        (y-intercept to green line);\r\n" );
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document.write( "    P2 = (7/3,10/3)   (intersection point of the red and green line);\r\n" );
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document.write( "    P3 = (61/9,0)     (x-intercept to red line).\r\n" );
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document.write( "You should calculate the value of the objective function Z = 5x + 30y  at these three points:\r\n" );
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document.write( "    at  P1:   Z = 5*0      + 30*8       = 240;\r\n" );
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document.write( "    at  P2:   Z = 5*(7/3)  + 30*(10/3)  = 111.667;\r\n" );
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document.write( "    at  P3:   Z = 5*(61/9) + 30*0       = 33.889.\r\n" );
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document.write( "The minimum is achieved at the point P3 = (61/9,0), where x= 61/9, y=0,  and is equal to 305/9 = 33.889 (approximately).\r\n" );
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\n" ); document.write( "\n" ); document.write( "To see other mini-max problems solved by the Linear Programming method, look into the lesson\r
\n" ); document.write( "\n" ); document.write( " - Solving minimax problems by the Linear Programming method \r
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