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document.write( "Consider the following linear program:
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document.write( "Min 2A + 2B
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document.write( "s.t.
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document.write( "1A + 3B ≤ 12
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document.write( "3A + 1B ≥ 13
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document.write( "1A - 1B ≥ 3
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document.write( "A, B ≥0
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document.write( "Change the problem to x's and y's by arbitrarily choosing\r\n" );
document.write( "x = A and y = B\r\n" );
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document.write( "Consider the following linear program:
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document.write( "Min 2x + 2y
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document.write( "s.t.
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document.write( "1x + 3y ≤ 12
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document.write( "3x + 1y ≥ 13
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document.write( "1x - 1y ≥ 3
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document.write( "x, y ≥ 0 \r
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document.write( "
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document.write( "a. Show the feasible region.
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document.write( "b. What are the extreme points of the feasible region?
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document.write( "c. Find the optimal solution using the graphical solution procedure.
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document.write( "The boundary line for 1x + 3y ≤ 12 has the equation which is\r\n" );
document.write( "like the inequality but has an equal sign in place of the\r\n" );
document.write( "inequality symbol. So the boundary line for 1x + 3y ≤ 12 has the\r\n" );
document.write( "equation 1x + 3y = 12. It has y-intercept (0,4) and x-intercept\r\n" );
document.write( "(12,0) and is the red line. Since the origin (0,0) substituted for\r\n" );
document.write( "(x,y) in the inequality 1x + 3y ≤ 12 gives 0 ≤ 12, the\r\n" );
document.write( "solution set for the inequality 1x + 3y ≤ 12 is ON THE SAME\r\n" );
document.write( "SIDE of the line as the origin is on, and since the origin is BELOW and\r\n" );
document.write( "LEFT of the RED line, the solution set is BELOW and LEFT of its boundary line, the RED LINE.\r\n" );
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document.write( "
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document.write( "---\r\n" );
document.write( "The boundary line for 3x + 1y ≥ 13 has the equation which is\r\n" );
document.write( "like the inequality but has an equal sign in place of the\r\n" );
document.write( "inequality symbol. So the boundary line for 3x + 1y ≥ 13 has the\r\n" );
document.write( "equation 3x + 1y = 13. It has y-intercept (0,13) and x-intercept\r\n" );
document.write( "(
,0) and is the GREEN line. Since the origin (0,0) substituted for\r\n" );
document.write( "(x,y) in the inequality 3x + 1y ≥ 13 gives 0 ≤ 13, which is\r\n" );
document.write( "FALSE, the solution set for the inequality 1x + 3y ≥ 12 is ON THE OPPOSITE\r\n" );
document.write( "SIDE of the line as the origin is on, and since the origin is BELOW and\r\n" );
document.write( "to the LEFT of the GREEN line, the solution set is ABOVE and to the RIGHT\r\n" );
document.write( "of its boundary line, the GREEN LINE.\r\n" );
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document.write( "
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document.write( "---\r\n" );
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document.write( "The boundary line for 1x - 1y ≥ 3 has the equation which is\r\n" );
document.write( "like the inequality but has an equal sign in place of the\r\n" );
document.write( "inequality symbol. So the boundary line for 1x - 1y ≥ 13 has the\r\n" );
document.write( "equation 1x - 1y = 3. It has y-intercept (0,3) and x-intercept\r\n" );
document.write( "(-3,0) and is the BLUE line. Since the origin (0,0) substituted for\r\n" );
document.write( "(x,y) in the inequality 1x - 1y ≥ 3 gives 0 ≤ 3, which is\r\n" );
document.write( "FALSE, the solution set for the inequality 1x - 1y ≥ 3 is ON THE OPPOSITE\r\n" );
document.write( "SIDE of the line as the origin is on, and since the origin is BELOW and\r\n" );
document.write( "to the LEFT of the BLUE line, the solution set is ABOVE and to the RIGHT\r\n" );
document.write( "of its boundary line, the BLUE LINE. \r\n" );
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document.write( "So the triangle indicated by\r\n" );
document.write( "F.R. is the Feasible Region.\r\n" );
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document.write( "
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document.write( "Now we must find the three corner points (vertices) of the triangle\r\n" );
document.write( "region \r\n" );
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document.write( "We solve the equations of the red and green lines by substitution\r\n" );
document.write( "or elimination and get x = 
, and y = 
. \r\n" );
document.write( "That's the point 
\r\n" );
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document.write( "\r\n" );
document.write( "We solve the equations of the red and blue lines by substitution\r\n" );
document.write( "or elimination and get x = 
, and y = 
. \r\n" );
document.write( "That's the point 
\r\n" );
document.write( "\r\n" );
document.write( "we solve the equations of the blue and green lines by substitution\r\n" );
document.write( "or elimination and get x = 4, and y = 1. \r\n" );
document.write( "That's the point (4,1)\r\n" );
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document.write( "Finally we calculate the objective expression 2x + 2y at each of\r\n" );
document.write( "those corner points:\r\n" );
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document.write( "Substituting x = , and y = into 2x + 2y gives 
.\r\n" );
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document.write( "Substituting x = , and y = into 2x + 2y gives 15.\r\n" );
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document.write( "Substituting x = 4, and y = 1 into 2x + 2y gives 10\r\n" );
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document.write( "So the minimum value of 2x + 2y is 10 when x = 4 and y = 1.\r\n" );
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document.write( "Edwin
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