document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199207>Question 1199207</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A company manufactures two products, A and B, and each of these products must be processed on two different machines. Product A requires 2 min of work time per unit on machine 1 and 4 min of work time per unit on machine 2. Product B requires 3 min of work time per unit on machine 1 and 1 min of work time per unit on machine 2. Each day 200 min are available on machine 1 and 300 min are available on machine2. To satisfy certain customers, the company must produce at least 8 units per day of product A and at least 10 units per day of product B. If the profit of each unit of product A is Php50 and the profit of each unit of product B is Php60, how many units of each product should be produced daily in order to maximize the company's profits?\r<BR>\n" );
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document.write( "Provide the following:\r<BR>\n" );
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document.write( "The objective function:   blank\r<BR>\n" );
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document.write( "The machine 1 constraint:  blank\r<BR>\n" );
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document.write( "The machine 2 constraint:  blank\r<BR>\n" );
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document.write( "The production constraints:  blank\r<BR>\n" );
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document.write( "The implicit constraints:  blank\r<BR>\n" );
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document.write( "The  number of product A  and product B  to give an optimum profit:  blank\r<BR>\n" );
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document.write( "The optimum cost: <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 #832968 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=832968\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The objective function...<br><BR>\n" );
document.write( "The profit is 50 for each product A and 60 for each product B:<BR>\n" );
document.write( "P = 50A+60B<br><BR>\n" );
document.write( "The machine 1 constraint...<br><BR>\n" );
document.write( "2 minutes for each product A and 3 minutes for each product B; 200 total minutes available:<BR>\n" );
document.write( "2A+3B <= 200<br><BR>\n" );
document.write( "The machine 2 constraint...<br><BR>\n" );
document.write( "4 minutes for each product A and 1 minute for each product B; 300 total minutes available:<BR>\n" );
document.write( "4A+B <= 300<br><BR>\n" );
document.write( "The production constraints...<br><BR>\n" );
document.write( "At least 8 of product A and at least 10 of product B:<BR>\n" );
document.write( "A >= 8<BR>\n" );
document.write( "B >= 10<br><BR>\n" );
document.write( "The implicit constraints...<br><BR>\n" );
document.write( "A >= 0; B >= 0<BR>\n" );
document.write( "(not needed, because the production constraints are more restrictive)<br><BR>\n" );
document.write( "The number of product A and product B to give an optimum profit...<br><BR>\n" );
document.write( "I'll leave it to you or another tutor to make a graph of the feasibility region and find the answer by whatever method.<br><BR>\n" );
document.write( "Note that the standard process described in virtually all references is NOT required.  The corner of the feasibility region where the objective function is maximized can be determined by comparing the slopes of the constraint boundary lines to the slope of the objective function; it is NOT necessary to evaluate the objective function at every corner of the feasibility region.<br><BR>\n" );
document.write( "The slopes of the constraint boundary lines are -4 and -2/3; the slope of the constraint function is -5/6.  Because the slope of the objective function is between the slopes of the constraint boundary lines, the objective function will be maximized where the constraint boundary lines intersect (as long as the production constraints are satisfied).<br><BR>\n" );
document.write( "2A+3B=200<BR>\n" );
document.write( "4A+B=300<BR>\n" );
document.write( "4A+6B=400<BR>\n" );
document.write( "5B=100<BR>\n" );
document.write( "B=20<BR>\n" );
document.write( "A=70<br><BR>\n" );
document.write( "That satisfies the production constraints, so<br><BR>\n" );
document.write( "ANSWER: The maximum profit is when 70 of product A and 20 of product B are produced.<br><BR>\n" );
document.write( "The optimum <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cross%28cost%29 ALIGN=MIDDLE  ALT=\"cross%28cost%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > profit:<BR>\n" );
document.write( "ANSWER: 70(50)+20(60) = 3500+1200 = 4700<br><BR>\n" );
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