document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1179568>Question 1179568</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi! can please help me with this question ? Thankyou!\r<BR>\n" );
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document.write( "The dean of Faculty of Management and Information Technology must plan the<BR>\n" );
document.write( "faculty’s course offerings for the first semester, 2020/2021. Student demands make it necessary to offer at least 30 undergraduate and 20 graduate courses in the term.Faculty contracts also dictate that at least 60 courses be offered in total. Each undergraduate course taught course the faculty an average of RM2,500 in faculty wages, and each graduate course costsRM3,000. By using the graphical method, calculate the number of undergraduate and graduate courses should be taught in the first semester so that the total faculty salaries are kept to a minimum. We let\r<BR>\n" );
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document.write( "U = undergraduate courses, and<BR>\n" );
document.write( "G = graduate courses.\r<BR>\n" );
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document.write( "Note: You might have to prepare a report for proposing an optimal solution by usingquantitative analysis techniques.\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 #850234 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52835)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850234\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "Hi! can please help me with this question ? Thankyou!\r<BR>\n" );
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document.write( "The dean of Faculty of Management and Information Technology must plan the faculty’s course <BR>\n" );
document.write( "offerings for the first semester, 2020/2021. Student demands make it necessary to offer <BR>\n" );
document.write( "at least 30 undergraduate and 20 graduate courses in the term. Faculty contracts <BR>\n" );
document.write( "also dictate that at least 60 courses be offered in total. Each undergraduate course <BR>\n" );
document.write( "taught course the faculty an average of RM2,500 in faculty wages, and each graduate course <BR>\n" );
document.write( "costs RM3,000. By using the graphical method, calculate the number of undergraduate <BR>\n" );
document.write( "and graduate courses should be taught in the first semester so that the total <BR>\n" );
document.write( "faculty salaries are kept to a minimum. We let\r<BR>\n" );
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document.write( "U = undergraduate courses, and<BR>\n" );
document.write( "G = graduate courses.\r<BR>\n" );
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document.write( "Note: You might have to prepare a report for proposing an optimal solution by using quantitative <BR>\n" );
document.write( "analysis techniques.<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Surely, this problem can be solved using Linear Programming method.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;But it also can be solved, &nbsp;using logical reasoning and common sense, \r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;practically mentally, &nbsp;in your mind.\r<BR>\n" );
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document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "First, notice that if the conditions \"at least 30 U and at least 20 G\" are satisfied\r\n" );
document.write( "and we have MORE than 60 courses, than the total number of courses can be diminished\r\n" );
document.write( "by 1 such way that we still will be in the feasible domain.\r\n" );
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document.write( "Indeed, if U >= 30 and G >= 20, and U + G > 60, it means that at least one number U or G\r\n" );
document.write( "is greater than their corresponding lower boundaries, so this course can be eliminated.\r\n" );
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document.write( "Repeating this reasoning as many times as required, we will get the condition\r\n" );
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document.write( "    U >= 30, G >= 20, U + G = 60  \r\n" );
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document.write( "so the total number of courses is precisely 60. Eliminating excessive courses, we diminish\r\n" );
document.write( "the total cost, so it is good, from the problem's point of view.\r\n" );
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document.write( "Now, if the number of graduate courses is G, then the number of undergraduate courses is U = 60-G.\r\n" );
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document.write( "The total cost is then  2500*U + 3000*G = 2500*(60 - G) + 3000*G = 150000 + 500*G,\r\n" );
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document.write( "and we want to make it as small as it is possible under restrictions.\r\n" );
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document.write( "From the last formula, it is clear that the total cost is minimum when the number of graduate \r\n" );
document.write( "courses G is at its lower bound G = 20.\r\n" );
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document.write( "Thus the optimal solution is G = 20 graduate courses and U = 60-20 = 40 undergraduate courses,\r\n" );
document.write( "giving the minimum possible cost  40*2500 + 20*3000 = 100000 + 60000 = RM160,000.\r\n" );
document.write( "</PRE>\r<BR>\n" );
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document.write( "Solved.\r<BR>\n" );
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document.write( "So, if the dean of Faculty of Management and Information Technology has a knack for logic,<BR>\n" );
document.write( "he can solve this problem MENTALLY, without using the LP-method.\r<BR>\n" );
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document.write( "The idea is to assign as few of expensive courses as possible under restrictions,<BR>\n" );
document.write( "and then to add as many of cheaper courses to get other restrictions.\r<BR>\n" );
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document.write( "It is another view to the problem, which is always good to have, if possible.\r<BR>\n" );
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