document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1202803>Question 1202803</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Sally Sethness assembles stereo equipment for resale in her shop. She offers two products, turntables and cassette players. She makes a profit of $10 on each turntable and $6 on each cassette. Both must go through two steps in her shop—assembly and bench checking. A turntable takes 12 hours to assemble and 4 hours to bench check. A cassette player takes 4 hours to assemble but 8 hours to bench check. Looking at this month's schedule, Sally sees that she has 60 assembly hours uncommitted and 40 hours of bench-checking time available. Use graphic linear programming to find her best combination of turntables and cassette players. What is the total profit on the combination you found? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #837919 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13209)</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=837919\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Tutor @ikleyn has provided a complete detailed description of the standard solution method, so look at her response for a good solution.<br><BR>\n" );
document.write( "However, there is a shortcut to the solution which can be a big time saver if the linear programming problem is more complex -- for example, a problem having three or even more constraints instead of the standard two.<br><BR>\n" );
document.write( "Virtually all resources on linear programming, including the link on this site she references in her response, say that to find the maximum or minimum value of the objective function you need to evaluate the function at all corners of the feasibility region.<br><BR>\n" );
document.write( "That is not true.<br><BR>\n" );
document.write( "The corner of the feasibility region where the objective function is maximum or minimum can be determined by comparing the slope of the objective function to the slopes of the constraint boundary lines.  Geometrically, what you are doing with this process is finding where a line with the slope of the objective function just touches the feasibility region.<br><BR>\n" );
document.write( "In this problem, the slopes of the constraint boundary lines are -1/2 and -3; the slope of the objective function is -5/3.  Since -5/3 is between -1/2 and -3, the objective function value will be maximized at the intersection of the two constraint boundary lines.<br><BR>\n" );
document.write( "Part of the standard process, a shown in her response, is to find the corners of the feasibility region, which involves solving pairs of equations of the constraint boundary lines.  In this problem, with two constraints, there is just one such intersection, so doing the algebra to find that single intersection is not much work.  But in a more complex problem with more constraints, it would be a lot of work to find all the intersection points.<br><BR>\n" );
document.write( "Using this method of comparing the slopes of the lines means you always know where the objective function will be maximized, so you only have to find one of the points of intersection of the constraint boundary lines.<br><BR>\n" );
document.write( "In this problem, the intersection of the two constraint boundary lines is at (4,3), so the objective function is maximized when she produces 4 turntables and 3 cassette players; the maximum profit is 4($10)+3($6) = $40+$18 = $58.<br><BR>\n" );
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