document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210545>Question 1210545</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Gorimapa Nigeria plc has just received an order for it's bathroom cabinet which is made up of two kinds that is standard and deluxe. The order is for at least 200 bathroom cabinets of either varieties and including at least 60 of the deluxe kind. The standard model takes 4 hours of the assembling time and has a valuable cost of #4000 whereas the deluxe model takes 5 hours of assembling time and has a valuable cost of #6000 . There are 400 hours available for assembling time. The equipment can be used to assemble either kind of cabinet in any combination. The company's manager engages you as an expert and wishes to minimize the company's cost of this special order. You are required as an expert to formulate this problem in a linear programming form and using the graphical method advise the manager on the best product that will enable his firm to minimize it's cost </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #853583 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2189)</B></A><img src=\"/images/stars/stars-13.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=853583\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To help Gorimapa Nigeria PLC minimize its costs while meeting the order requirements, we can use **Linear Programming (LP)**. This mathematical approach allows us to find the most efficient combination of standard and deluxe cabinets within the given constraints.\r<BR>\n" );
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document.write( "### 1. Problem Formulation\r<BR>\n" );
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document.write( "First, we define our variables and the mathematical relationships between them.\r<BR>\n" );
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document.write( "**Decision Variables:**\r<BR>\n" );
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document.write( "* Let  = Number of **Standard** cabinets to produce.<BR>\n" );
document.write( "* Let  = Number of **Deluxe** cabinets to produce.\r<BR>\n" );
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document.write( "**Objective Function:**<BR>\n" );
document.write( "The goal is to minimize the total variable cost ():\r<BR>\n" );
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document.write( "**Constraints:**\r<BR>\n" );
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document.write( "1. **Total Order Quantity:** At least 200 cabinets in total.<BR>\n" );
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document.write( "2. **Deluxe Requirement:** At least 60 deluxe cabinets must be made.<BR>\n" );
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document.write( "3. **Assembly Time:** Total time cannot exceed 800 hours.<BR>\n" );
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document.write( "4. **Non-negativity:** Production cannot be negative.<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### 2. Graphical Method Analysis\r<BR>\n" );
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document.write( "To solve this graphically, we treat the inequalities as equations to find the boundary lines.\r<BR>\n" );
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document.write( "* **Line 1 (Total Order):** .<BR>\n" );
document.write( "* If ; if .\r<BR>\n" );
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document.write( "* **Line 2 (Deluxe Min):** .<BR>\n" );
document.write( "* A horizontal line at .\r<BR>\n" );
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document.write( "* **Line 3 (Time Limit):** .<BR>\n" );
document.write( "* If ; if .\r<BR>\n" );
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document.write( "#### Identifying the Feasible Region\r<BR>\n" );
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document.write( "The feasible region is the area that satisfies all three conditions simultaneously. Looking at the intercepts:\r<BR>\n" );
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document.write( "* The **Time Limit** and **Total Order** lines actually meet at the x-axis at .<BR>\n" );
document.write( "* However, we must have at least 60 Deluxe cabinets ().\r<BR>\n" );
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document.write( "By solving the intersection of  and the other constraints:\r<BR>\n" );
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document.write( "1. **Intersection of  and :** . Point: **(140, 60)**.<BR>\n" );
document.write( "2. **Intersection of  and :** . Point: **(125, 60)**.\r<BR>\n" );
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document.write( "**Wait!** Note that at point (140, 60), the total hours needed would be  hours. This exceeds our 800-hour limit.\r<BR>\n" );
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document.write( "### 3. Finding the Optimal Solution\r<BR>\n" );
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document.write( "The feasible region is a small triangle bounded by the points where these constraints overlap. To minimize cost, we test the corner points of the feasible region:\r<BR>\n" );
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document.write( "| Corner Point  | Cost Calculation () | Total Cost |<BR>\n" );
document.write( "| --- | --- | --- |<BR>\n" );
document.write( "| **(125, 60)** |  | **#860,000** |<BR>\n" );
document.write( "| **(0, 160)** |  | #960,000 |<BR>\n" );
document.write( "| **(0, 200)** | *Not feasible (Exceeds 800 hours)* | - |\r<BR>\n" );
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document.write( "*Note: The point (200, 0) is also not feasible because  must be at least 60.*\r<BR>\n" );
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document.write( "### 4. Expert Advice to the Manager\r<BR>\n" );
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document.write( "Based on the linear programming analysis, the best product mix to minimize costs while staying within the 800-hour limit and meeting the order requirements is:\r<BR>\n" );
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document.write( "* **Standard Cabinets:** 125 units<BR>\n" );
document.write( "* **Deluxe Cabinets:** 60 units<BR>\n" );
document.write( "* **Total Production:** 185 units<BR>\n" );
document.write( " </SPAN>\n" );
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