document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=835823>Question 835823</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Mr. Subhash has taken Rs. 100,000 from his father to invest them in a combination of only two stock portfolios with the maximum investment allowed in either portfolio set at Rs. 75,000. The first portfolio has an average return of 10% whereas the second has 20%. In terms of risk factors associated with these portfolios, the first has a risk rating of 4 (on a scale from 0 to 10), and the second has 9. Since he wants to maximize his return, he will not accept an average rate return below 12% or a risk above 6. Hence, he then faces the important question. How much should he invest in each portfolio?<BR>\n" );
document.write( "(a)Formulate the above as linear programming problem.<BR>\n" );
document.write( "(b)Solve (a) by graphical method.<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 #503784 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=503784\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> let x equal the amount of money to invest at 10%.<BR>\n" );
document.write( "let y equal the amount of money to invest at 20%.<BR>\n" );
document.write( "your objective function will be to maximize your return.<BR>\n" );
document.write( "this function is therefore:<BR>\n" );
document.write( "return = .10 * x + .20 * y<BR>\n" );
document.write( "your constraint equations will be as follows:<BR>\n" );
document.write( "x >= 0<BR>\n" );
document.write( "y >= 0<BR>\n" );
document.write( "this is because the amount invested can't be less than 0.<BR>\n" );
document.write( "x + y <= 100,000<BR>\n" );
document.write( "this is because the total amount to be invested cannot exceed 100,000.<BR>\n" );
document.write( "x <= 75<BR>\n" );
document.write( "y <= 75<BR>\n" );
document.write( "this is because the maximum investment in either portfolio cannot be greater than 75,000.<BR>\n" );
document.write( "4x + 9y <= 6(x+y)<BR>\n" );
document.write( "this is because the average risk factor has to be less than or equal to 6.<BR>\n" );
document.write( ".10x + .20y >= .12(x+y)<BR>\n" );
document.write( "this is because the average rate of return cannot be less than 12%.\r<BR>\n" );
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document.write( "the problem has been formulated.<BR>\n" );
document.write( "what is left is to graph and find the solution.\r<BR>\n" );
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document.write( "the graph of all these equations is shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2014/jan313.jpg\" alt = \"$$$$\" < />\r<BR>\n" );
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document.write( "your feasible region is the shaded area on or below the line of 4x + 9y <= 6(x+y) and on or below the line of x + y <= 100 and on or to the left of the line of x <= 75 and on or above the line of .10x + .20y >= .12 * (x+y).\r<BR>\n" );
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document.write( "this area satisfies all the constraints.<BR>\n" );
document.write( "the maximum return will be at the intersection points of the lines bounding this feasible region.\r<BR>\n" );
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document.write( "those points, and the value of the objective function at those points, are:<BR>\n" );
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document.write( "(0,0)          0 * .1 + 0 * .2 = 0\r\n" );
document.write( "(60,40)        60 * .1 + 40 * .2 = 14 ***** maximum return\r\n" );
document.write( "75,25)         75 * .1 + 25 * .2 = 12.5\r\n" );
document.write( "75,18.75)      75 * .1 + 18.75 * .2 = 11.25\r\n" );
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document.write( "the maximum return is when you invest 60,000 at 10% and 40,000 at 20%.<BR>\n" );
document.write( "this meets all the constraints of the project.<BR>\n" );
document.write( "maximum investment in each portfolio is less than 75,000<BR>\n" );
document.write( "all of the original 100,000 has been invested (requirement was less than or equal to 100,000).<BR>\n" );
document.write( "the average rate of return is greater than 12%.<BR>\n" );
document.write( "the average risk factor is less than or equal to 6.<BR>\n" );
document.write( "it is actually right on 6.  60 * 4 + 40 * 9 = 600 / 100 equals an average risk factor of 6.\r<BR>\n" );
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document.write( "the graph shows the original constraint equations and also the same equations after solving for y in each of the ones that i thought would be useful to show that way.\r<BR>\n" );
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