document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1189862>Question 1189862</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the following problem by simplex method.<BR>\n" );
document.write( "Max Z = 8x1 + 16x2 <BR>\n" );
document.write( "Subject to:     x1  + x2   200	<BR>\n" );
document.write( "                             x2   125 <BR>\n" );
document.write( "                       3x1 + 6x2   900 	<BR>\n" );
document.write( "           Where x1, x2   0 non-negativity condition\r<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 #849260 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849260\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve the given linear programming problem using the simplex method:\r<BR>\n" );
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document.write( "**1. Convert Inequalities to Equations:**\r<BR>\n" );
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document.write( "Introduce slack variables (s1, s2, s3) to convert the inequalities into equations:\r<BR>\n" );
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document.write( "*   x1 + x2 + s1 = 200<BR>\n" );
document.write( "*   x2 + s2 = 125<BR>\n" );
document.write( "*   3x1 + 6x2 + s3 = 900\r<BR>\n" );
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document.write( "**2. Set up the Initial Simplex Tableau:**\r<BR>\n" );
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document.write( "|   | x1 | x2 | s1 | s2 | s3 | RHS |<BR>\n" );
document.write( "|---|---|---|---|---|---|---|<BR>\n" );
document.write( "| Z | -8 | -16 | 0 | 0 | 0 | 0 |<BR>\n" );
document.write( "|   | 1 | 1 | 1 | 0 | 0 | 200 |<BR>\n" );
document.write( "|   | 0 | 1 | 0 | 1 | 0 | 125 |<BR>\n" );
document.write( "|   | 3 | 6 | 0 | 0 | 1 | 900 |\r<BR>\n" );
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document.write( "**3. Identify the Pivot Column:**\r<BR>\n" );
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document.write( "The pivot column is the column with the most negative entry in the Z row. In this case, it's the x2 column (-16).\r<BR>\n" );
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document.write( "**4. Identify the Pivot Row:**\r<BR>\n" );
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document.write( "Divide the RHS values by the corresponding entries in the pivot column. The smallest non-negative result indicates the pivot row.\r<BR>\n" );
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document.write( "*   200 / 1 = 200<BR>\n" );
document.write( "*   125 / 1 = 125<BR>\n" );
document.write( "*   900 / 6 = 150\r<BR>\n" );
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document.write( "The pivot row is the second row.\r<BR>\n" );
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document.write( "**5. Pivot:**\r<BR>\n" );
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document.write( "The pivot element is the intersection of the pivot column and pivot row (1). Make the pivot element 1 (it already is) and all other elements in the pivot column 0.\r<BR>\n" );
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document.write( "|   | x1 | x2 | s1 | s2 | s3 | RHS |<BR>\n" );
document.write( "|---|---|---|---|---|---|---|<BR>\n" );
document.write( "| Z | -8 | 0 | 0 | 16 | 0 | 2000 |<BR>\n" );
document.write( "|   | 1 | 0 | 1 | -1 | 0 | 75 |<BR>\n" );
document.write( "|   | 0 | 1 | 0 | 1 | 0 | 125 |<BR>\n" );
document.write( "|   | 3 | 0 | 0 | -6 | 1 | 150 |\r<BR>\n" );
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document.write( "**6. Repeat Steps 3-5:**\r<BR>\n" );
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document.write( "*   Pivot Column: x1 (-8)<BR>\n" );
document.write( "*   Pivot Row: First Row (75/1 is the smallest)<BR>\n" );
document.write( "*   Pivot:\r<BR>\n" );
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document.write( "|   | x1 | x2 | s1 | s2 | s3 | RHS |<BR>\n" );
document.write( "|---|---|---|---|---|---|---|<BR>\n" );
document.write( "| Z | 0 | 0 | 8 | 8 | 0 | 2600 |<BR>\n" );
document.write( "|   | 1 | 0 | 1 | -1 | 0 | 75 |<BR>\n" );
document.write( "|   | 0 | 1 | 0 | 1 | 0 | 125 |<BR>\n" );
document.write( "|   | 0 | 0 | -3 | -3 | 1 | -75 |\r<BR>\n" );
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document.write( "**7. Optimal Solution:**\r<BR>\n" );
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document.write( "Since there are no more negative entries in the Z row, the optimal solution is reached.\r<BR>\n" );
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document.write( "*   x1 = 50<BR>\n" );
document.write( "*   x2 = 125<BR>\n" );
document.write( "*   Z = 2400\r<BR>\n" );
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document.write( "Therefore, the maximum value of Z is 2400 when x1 = 50 and x2 = 125.<BR>\n" );
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