document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=532189>Question 532189</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider the following LP problem:<BR>\n" );
document.write( "maximum profit = 5X+6Y<BR>\n" );
document.write( "subject to       2X+y <- 120<BR>\n" );
document.write( "                 2x+3y <- 120<BR>\n" );
document.write( "                 x,y<- 0<BR>\n" );
document.write( "(a) What is the optimal solution to this problem?<BR>\n" );
document.write( "Solve it graphically.<BR>\n" );
document.write( "(b) If a technical breakthrough occurred that raised<BR>\n" );
document.write( "the profit per unit of X to $8, would this affect<BR>\n" );
document.write( "the optimal solution?<BR>\n" );
document.write( "(c) Instead of an increase in the profit coefficient X<BR>\n" );
document.write( "to $8, suppose that profit was overestimated and<BR>\n" );
document.write( "should only have been $3. Does this change the<BR>\n" );
document.write( "optimal solution?<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 #350712 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=350712\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "First let's get your terminology straight.  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large Z\ =\ 5x\ +\ 6y\"> is NOT the maximum profit.  It is the profit when <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> items having a profit of 5 each and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\"> items having a profit of 6 each are sold.  Your goal is to maximize this profit amount.\r<BR>\n" );
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document.write( "Second, don't mix capital letters and lower case letters for your variables (<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large X\"> for example).  They are different variables and stand for different quantities.  And I'm sure that is NOT what you meant when you wrote out the problem.\r<BR>\n" );
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document.write( "Part a).  Graph all four of your constraint inequalities: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 2x\ +\ y\ \leq\ 120\">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 2x\ +\ 3y\ \leq\ 120\">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ \geq\ 0\">, and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\ \geq\ 0\">.  Hint:  Ordinarily when you graph an inequality you shade in the half-plane that represents the solution set.  If you do that here, your area of feasibility will be that area where the four shaded areas (one each from each of your constraint inequalities) overlap.  However, if you do the shading using the opposite sense, that is shade in that part of the plane NOT in the solution set, then the area of feasibility ends up being totally unshaded and therefore much easier for visual comprehension.\r<BR>\n" );
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document.write( "Either way, the optimum solution, since none of the slopes of the constraint boundaries are equal to the slope of the function you are trying to optimize, you know that the optimum solution will be one of the four vertices of the feasibility quadrilateral.\r<BR>\n" );
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document.write( "So, find the four ordered pairs that represent the four vertices of the feasibility quadrilateral and evaluate the profit function for the values of the coordinates of each of those points.  The optimum solution will make the function have the larger value.\r<BR>\n" );
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document.write( "Part b) and c) Change your profit function to <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large Z_b\ =\ 8x\ +\ 6y\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large Z_c\ =\ 3x\ +\ 6y\">.  Then evaluate each of these new profit functions using the same potential optimum points as you used for part a.  See if the change in the profit function causes the optimum point to change.\r<BR>\n" );
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document.write( "Write back and we can negotiate a price for a full graphical solution if you are interested.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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