document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192581>Question 1192581</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the following linear programming problems using simplex method. <BR>\n" );
document.write( "Maximize Z = 3x+4y <BR>\n" );
document.write( "subject x+y <= 4, 2x+y <= 5, x >= 0,y >= 0. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #824460 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52781)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=824460\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "Solve the following linear programming problems using simplex method.<BR>\n" );
document.write( "Maximize Z = 3x+4y<BR>\n" );
document.write( "subject x+y <= 4, 2x+y <= 5, x >= 0,y >= 0.<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;I see a great progress on your side &nbsp;(&nbsp;!&nbsp;)\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;I see that after my criticism, &nbsp;you learned on how to formulate the problem correctly &nbsp;(&nbsp;!&nbsp;)\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It means that I can work with you and teach you the subject.\r<BR>\n" );
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document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "In coordinate plane, construct the feasibility domain for the given constraints.\r\n" );
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document.write( "So, in the first quadrant, you plot the lines y = 4-x  and y = 5-2x.\r\n" );
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document.write( "They are shown below in the Figure as red line and green line, respectively.\r\n" );
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document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+400%2C+400%2C+-1%2C+5%2C+-1%2C+5%2C%0D%0A++++++++++4-x%2C++5-2x%0D%0A%29 ALIGN=MIDDLE  ALT=\"graph%28+400%2C+400%2C+-1%2C+5%2C+-1%2C+5%2C%0D%0A++++++++++4-x%2C++5-2x%0D%0A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r\n" );
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document.write( "        The lines  y = 4-x (red)  and   y = 5-2x (green).\r\n" );
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document.write( "The feasibility domain is the quadrilateral in QI, which is below these lines.\r\n" );
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document.write( "It is the area, where you look and search for the maximum of the objective function.\r\n" );
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document.write( "The quadrilateral has vertices \r\n" );
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document.write( "    P0 = (0,0)    - the origin of the coordinate system;\r\n" );
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document.write( "    P1 = (0,4)    - y-intercept of the red line;\r\n" );
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document.write( "    P2 = (1,3)    - intersection of red line and green line;\r\n" );
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document.write( "    P3 = (2.5,0)  - x-intercept of the green line.\r\n" );
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document.write( "The given objective function is a linear function of x and y over the feasibility quadrilateral.\r\n" );
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document.write( "It it obvious that it gets its maximum value in some of the four vertices.\r\n" );
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document.write( "So you check the value of the objective fubction in vertices and find one of them where it has the maximum.\r\n" );
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document.write( "See the Table below, which I prepared for you:\r\n" );
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document.write( "    Point     coordinates   calculation of the      the value of the\r\n" );
document.write( "                 (x,y)      objective function      objective function\r\n" );
document.write( "                            Z(x,y) = 3x + 4y        at the vertex\r\n" );
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document.write( "     P0         (0,0)       3*0 + 4*0                    0\r\n" );
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document.write( "     P1         (0,4)       3*0 + 4*4                   16\r\n" );
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document.write( "     P2         (1,3)       3*1 + 4*3                   15\r\n" );
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document.write( "     P3         (2.5,0)     3*2.5 + 4*0                  7.5\r\n" );
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document.write( "From the table, you see that the objective function has the maximum value in vertex P1.\r\n" );
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document.write( "It means that the objective function has the maximum at the point P(0,4) at x= 0, y= 4.\r\n" );
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document.write( "<U>ANSWER</U>.  The solution to the given LP problem is the point P(0,4) at x= 0, y= 4.\r\n" );
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document.write( "         The maximum value is 16.\r\n" );
document.write( "</PRE>\r<BR>\n" );
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document.write( "Solved and thoroughly explained.\r<BR>\n" );
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document.write( "============\r<BR>\n" );
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document.write( "To see other similar problems solved by the Linear Programming method, &nbsp;look into the lesson\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/word/misc/Solving-minimax-problems-by--the-Linear-Programming-method.lesson>Solving minimax problems by the Linear Programming method</A> \r<BR>\n" );
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document.write( "in this site.\r<BR>\n" );
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document.write( "Also, &nbsp;look into the solutions of other similar problems in the archive to this forum under the links\r<BR>\n" );
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document.write( "https://www.algebra.com/algebra/homework/Linear-equations/Linear-equations.faq.question.1148776.html\r<BR>\n" );
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document.write( "https://www.algebra.com/algebra/homework/word/misc/Miscellaneous_Word_Problems.faq.question.1142297.html \r<BR>\n" );
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document.write( "https://www.algebra.com/algebra/homework/Inequalities/Inequalities.faq.question.1142000.html \r<BR>\n" );
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document.write( "https://www.algebra.com/algebra/homework/Inequalities/Inequalities.faq.question.1141781.html  \r<BR>\n" );
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document.write( "https://www.algebra.com/algebra/homework/Sequences-and-series/Sequences-and-series.faq.question.1137172.html\r<BR>\n" );
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document.write( "https://www.algebra.com/algebra/homework/Graphs/Graphs.faq.question.1136382.html\r<BR>\n" );
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document.write( "https://www.algebra.com/algebra/homework/Coordinate-system/Coordinate-system.faq.question.1134444.html\r<BR>\n" );
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document.write( "https://www.algebra.com/algebra/homework/Graphs/Graphs.faq.question.1131906.html\r<BR>\n" );
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document.write( "https://www.algebra.com/algebra/homework/coordinate/word/Linear_Equations_And_Systems_Word_Problems.faq.question.1131043.html\r<BR>\n" );
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document.write( "https://www.algebra.com/algebra/homework/word/finance/Money_Word_Problems.faq.question.1129285.html\r<BR>\n" );
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document.write( "https://www.algebra.com/algebra/homework/Finance/Finance.faq.question.1128383.html  \r<BR>\n" );
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document.write( "https://www.algebra.com/algebra/homework/Linear-equations/Linear-equations.faq.question.1123217.html   \r<BR>\n" );
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document.write( "https://www.algebra.com/algebra/homework/playground/test.faq.question.1112482.html\r<BR>\n" );
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document.write( "https://www.algebra.com/algebra/homework/Finance/Finance.faq.question.1102103.html\r<BR>\n" );
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document.write( " </SPAN>\n" );
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