document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1155591>Question 1155591</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A toymaker makes soldiers and guns from wood and plastic. The store has 60 units of wood and 44 units of plastic. Each soldier requires 10 units of wood and 4 units of plastic, whereas each gun requires 6 units of plastic and 4 units of wood. The store has to use a minimum of 22 units of plastic. The demand for soldiers is no more than 2. Each soldier is sold along with at least two guns. A soldier and a gun earn $30 and $20 in profit, respectively. Formulate a linear programming model for this problem with an appropriate objective function. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #778437 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=778437\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Let x = number of soldiers<BR>\n" );
document.write( "Let y = number of guns<br><BR>\n" );
document.write( "The constraints are....<BR>\n" );
document.write( "(1) 10x+4y <= 60  (maximum units of wood)<BR>\n" );
document.write( "(2) 22 <= 4x+6y <= 44  {minimum and maximum units of plastic)<BR>\n" );
document.write( "(3) x <= 2  (maximum demand for soldiers)<BR>\n" );
document.write( "(4) y >= 2x  (at least 2 guns for each soldier)<br><BR>\n" );
document.write( "and, of course, x>=0 and y>=0<br><BR>\n" );
document.write( "Sketch a graph of the constraint boundary lines and the resulting feasibility region.<br><BR>\n" );
document.write( "Contrary to what is usually taught, it is NOT necessary to evaluate the objective function at every corner of the feasibility region.<br><BR>\n" );
document.write( "Instead, the corner at which the objective function is maximized can be determined by comparing the slopes of the constraint boundary lines and the objective function.<br><BR>\n" );
document.write( "The objective function is P = 30x+20y; its slope is -3/2.  The maximum value of the objective function will be obtained at the corner of the feasibility region where a line with slope -3/2 just touches the feasibility region.<br><BR>\n" );
document.write( "If you have drawn your graph carefully and know the slopes of the constraint boundary lines, you will see that happens at the intersection of the constraint boundary lines x=2 and 4x+6y=44.  The coordinates of that corner are (2,6).<br><BR>\n" );
document.write( "The value of the objective function at that corner is 30(2)+20(6) = 60+120 = 180.<br><BR>\n" );
document.write( "ANSWERS:  The maximum profit is $180, when 2 soldiers and 6 guns are produced.<br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );