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 #850218 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850218\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Define the Objective Function**\r<BR>\n" );
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document.write( "The objective is to minimize total faculty salaries.  \r<BR>\n" );
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document.write( "* Cost of undergraduate courses: RM2500 per course<BR>\n" );
document.write( "* Cost of graduate courses: RM3000 per course\r<BR>\n" );
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document.write( "Therefore, the objective function is:\r<BR>\n" );
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document.write( "**Minimize Z = 2500U + 3000G** where:<BR>\n" );
document.write( "* Z = Total faculty salaries<BR>\n" );
document.write( "* U = Number of undergraduate courses<BR>\n" );
document.write( "* G = Number of graduate courses\r<BR>\n" );
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document.write( "**2. Formulate the Constraints**\r<BR>\n" );
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document.write( "* **Student Demand:**<BR>\n" );
document.write( "    * At least 30 undergraduate courses: U ≥ 30<BR>\n" );
document.write( "    * At least 20 graduate courses: G ≥ 20\r<BR>\n" );
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document.write( "* **Faculty Contracts:**<BR>\n" );
document.write( "    * At least 60 total courses: U + G ≥ 60\r<BR>\n" );
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document.write( "* **Non-negativity:**<BR>\n" );
document.write( "    * U ≥ 0 <BR>\n" );
document.write( "    * G ≥ 0 (You can't have negative courses)\r<BR>\n" );
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document.write( "**3. Graphical Solution**\r<BR>\n" );
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document.write( "* **Plot the Constraints:**<BR>\n" );
document.write( "    1. Treat each inequality as an equation and plot the lines on a graph with U on the x-axis and G on the y-axis.<BR>\n" );
document.write( "    2. For U ≥ 30, plot the vertical line U = 30 and shade the region to the right.<BR>\n" );
document.write( "    3. For G ≥ 20, plot the horizontal line G = 20 and shade the region above.<BR>\n" );
document.write( "    4. For U + G ≥ 60, plot the line U + G = 60 (it intersects the axes at U=60 and G=60) and shade the region above the line.\r<BR>\n" );
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document.write( "* **Identify the Feasible Region:** The feasible region is the area where all shaded regions overlap. This represents all the combinations of U and G that satisfy the constraints.\r<BR>\n" );
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document.write( "* **Find the Corner Points:** The feasible region will be a polygon.  Identify the coordinates of each corner point of this polygon.\r<BR>\n" );
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document.write( "* **Evaluate the Objective Function:** Substitute the U and G values of each corner point into the objective function (Z = 2500U + 3000G).  The corner point that gives the minimum value for Z is the optimal solution.\r<BR>\n" );
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document.write( "**4. Interpretation and Report**\r<BR>\n" );
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document.write( "Let's assume you've done the graphing and found the optimal solution. Here's how you might structure a report:\r<BR>\n" );
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document.write( "**Report: Optimization of Faculty Course Offerings**\r<BR>\n" );
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document.write( "**Objective:**\r<BR>\n" );
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document.write( "To determine the optimal number of undergraduate (U) and graduate (G) courses to offer in the first semester of 2020/2021, minimizing total faculty salary costs while meeting student demand and faculty contract requirements.\r<BR>\n" );
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document.write( "**Methodology:**\r<BR>\n" );
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document.write( "Linear programming with a graphical solution approach was used. The objective function and constraints were defined as follows:\r<BR>\n" );
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document.write( "* **Objective Function:** Minimize Z = 2500U + 3000G<BR>\n" );
document.write( "* **Constraints:**<BR>\n" );
document.write( "    * U ≥ 30<BR>\n" );
document.write( "    * G ≥ 20<BR>\n" );
document.write( "    * U + G ≥ 60<BR>\n" );
document.write( "    * U, G ≥ 0\r<BR>\n" );
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document.write( "The feasible region was graphically determined, and the objective function was evaluated at each corner point.\r<BR>\n" );
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document.write( "**Results:**\r<BR>\n" );
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document.write( "*(Here, you would state the optimal solution you found graphically, for example: U = 40, G = 20)*\r<BR>\n" );
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document.write( "The optimal solution is to offer 40 undergraduate courses and 20 graduate courses. This will minimize total faculty salaries to RM [insert calculated minimum cost].\r<BR>\n" );
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document.write( "**Conclusion:**\r<BR>\n" );
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document.write( "By offering 40 undergraduate and 20 graduate courses, the Faculty of Management and Information Technology can meet student demand, fulfill faculty contract obligations, and minimize salary costs.\r<BR>\n" );
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document.write( "**Recommendations:**\r<BR>\n" );
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document.write( "* The faculty should plan its course offerings accordingly.<BR>\n" );
document.write( "* This analysis can be revisited if student demand or faculty contract requirements change.<BR>\n" );
document.write( "* Further analysis could incorporate other factors, such as classroom availability and resource allocation. </SPAN>\n" );
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