document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1118359>Question 1118359</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me to understand how to do \"optimization\"\r<BR>\n" );
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document.write( "Objective function: C = 8x+10y Maximize<BR>\n" );
document.write( "Constraints<BR>\n" );
document.write( "2x+y _< 12<BR>\n" );
document.write( "x+3y _< 21\r<BR>\n" );
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document.write( "X_>0, y_>0\r<BR>\n" );
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document.write( "Then I need to graph it?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #733650 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=733650\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Every resource I have ever seen explaining how to solve this kind of problem says that you have to find the value of the objective function at every corner of the feasibility region.  That is simply not true.<br><BR>\n" );
document.write( "Let's look again at your problem, where the constraint functions are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3C=-2x%2B12 ALIGN=MIDDLE  ALT=\"y%3C=-2x%2B12\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3C=%28-1%2F3%29x%2B7 ALIGN=MIDDLE  ALT=\"y%3C=%28-1%2F3%29x%2B7\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-2%2C14%2C-2%2C14%2C-2x%2B12%2C%28-1%2F3%29x%2B7%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-2%2C14%2C-2%2C14%2C-2x%2B12%2C%28-1%2F3%29x%2B7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The objective function is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C+=+8x%2B10y ALIGN=MIDDLE  ALT=\"C+=+8x%2B10y\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "In slope-intercept form, that is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28-8%2F10%29x%2BC%2F10 ALIGN=MIDDLE  ALT=\"y+=+%28-8%2F10%29x%2BC%2F10\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "We don't know the y-intercept, because we don't know the value of C.  But we know the slope of the objective function is -8/10, or -4/5.<br<><BR>\n" );
document.write( "Then to find the maximum value of the objective function you only need to determine which corner point of the feasibility region will be just touched by a line with slope -4/5.  And since that slope -4/5 is between the slopes of the two constraint functions (-2 and -1/3), the corner point you want is at the intersection of the two constraint lines.<br><BR>\n" );
document.write( "After that, you find that the intersection point is (3,6), and you evaluate the objective function at that point only, finding that the maximum value of the objective function is 10(3)+8(6) = 30+48 = 76.<br><BR>\n" );
document.write( "If the slope of the objective function had been -3, which is smaller than -2, then the maximum value would have been obtained at (6,0); if the slope of the objective function had been -1/5, which is larger than -1/3, the maximum value would have been obtained at (0,12). </SPAN>\n" );
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