document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1155269>Question 1155269</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A manufacturer of wooden articles produces tables and chairs which require two types of inputs mainly, the being wood and labour. The manufacturer knows that for a table 3 units of wood and 1 unit of labour are required while for a chair they are 2 units each. The profit from each table is 20 USD while it is 16 USD for each chair. The total available resources for the manufacture are 150 units of wood and 75 units of labour. The minimum demand for chairs is 10 and the minimum demand for tables is 8.  The manufacturer wants to maximize his profit by distributing his resources for tables and chairs. Formulate the problem mathematically. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #777862 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=777862\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> this is a linear programming type problem.<BR>\n" );
document.write( "x represents the number of tables.<BR>\n" );
document.write( "y represents the number of chair.\r<BR>\n" );
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document.write( "the objective function is profit = 20 * x + 16 * y\r<BR>\n" );
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document.write( "the constraints are:\r<BR>\n" );
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document.write( "3 * x + 2 * y <= 150<BR>\n" );
document.write( "that's 3 units of wood for * the number of tables plus 2 units of wood * the number of chairs.\r<BR>\n" );
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document.write( "x + 2 * y <= 75<BR>\n" );
document.write( "that's 1 unit of labor * the number of tables plus 2 units of labor * the number of chair.\r<BR>\n" );
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document.write( "x >= 8<BR>\n" );
document.write( "that's because the minimum number of tables is 8.\r<BR>\n" );
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document.write( "y >= 10<BR>\n" );
document.write( "that's because the minimum number of chairs is 10.\r<BR>\n" );
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document.write( "x,y >= 0<BR>\n" );
document.write( "that's because the minimum number of chair or tables is equal to 0, i.e. it can't be negative.\r<BR>\n" );
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document.write( "if you drew a table, it might make it easier to understand what's going on.\r<BR>\n" );
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document.write( "the table might look like this.\r<BR>\n" );
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document.write( "                      tables           chairs         total\r\n" );
document.write( "number of               x                 y\r\n" );
document.write( "units of wood           3                 2           <= 150\r\n" );
document.write( "units of labor          1                 2           <= 75\r\n" );
document.write( "demand                  >= 8              >=  10\r\n" );
document.write( "profit                  20                16\r\n" );
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document.write( "to summarize the formulas required, you get:\r<BR>\n" );
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document.write( "objective function is profit = 20x + 16y<BR>\n" );
document.write( "units of wood constraint is 3x + 2y <= 150<BR>\n" );
document.write( "labor constraint is x + 2y <= 75<BR>\n" );
document.write( "demand constraint is x >= 8 and y >= 10<BR>\n" );
document.write( "profit = 20x + 16y<BR>\n" );
document.write( "no negative units is x >= 0 and y >= 0<BR>\n" );
document.write( "this requirement is redundant in this problem because you already have x >= 8 and y >= 10 which eliminates the need for x >= 0 and y >= 0.   this will not always be the case, depending on the problem.\r<BR>\n" );
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document.write( "there is a simplex method tool that can be used to solve a problem such as this.\r<BR>\n" );
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document.write( "that tool can be found at <a  rel=nofollow HREF= \"\" target = \"_blank\"></a>\r<BR>\n" );
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document.write( "the output of that tool looks like this.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2020/040402.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "the maximum profit is 1050.<BR>\n" );
document.write( "37.5 units of wood are used and 18.75 units of labor are used.\r<BR>\n" );
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document.write( "the graphical solution confirms the answer.\r<BR>\n" );
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document.write( "it looks like this.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2020/040403.jpg\" alt=\"%%%\" >\r<BR>\n" );
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document.write( "with the desmos.com calculator, you plot the opposite of your constraints.<BR>\n" );
document.write( "the area of the graph that is not shaded is you region of feasibility.<BR>\n" );
document.write( "the corner points of that region are where your maximum profit will be.<BR>\n" );
document.write( "you evaluate your profit function at each corner point.<BR>\n" );
document.write( "for example, when x = 37.5 and y = 18.75, your profit is 20 * 37.5 + 16 * 18.75 = 1050.\r<BR>\n" );
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document.write( "the simplex tool also contains a tutorial on how to manually develop a solution using the simplex method.\r<BR>\n" );
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document.write( "the graphing method can be used with two variables.<BR>\n" );
document.write( "anything more than that uses different methods that are not graphical, such as the simplex method and others.\r<BR>\n" );
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document.write( "excel can also be used but it's more difficult to set up than the simplex method tool i showed you.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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