document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1171735>Question 1171735</A>: <I><SPAN STYLE=\"word-break: break-all;\">  The corporate and strategic department has decided to add exactly 200 km of<BR>\n" );
document.write( "highway and exactly 100 km of expressway to its road system this year. The<BR>\n" );
document.write( "standard price for road construction is RM1 million per kilometer of highway and<BR>\n" );
document.write( "RM5 million per kilometer of expressway. Only two contractors, company ABC and<BR>\n" );
document.write( "company RST, are eligible to proceed with the construction, so the entire 300 km of<BR>\n" );
document.write( "road must be built by these two companies.<BR>\n" );
document.write( "However, company ABC can construct at most 200 km of roadway (highway and<BR>\n" );
document.write( "expressway) and company RST can construct at most 150 km of roadway (highway<BR>\n" );
document.write( "and expressway). Each company must be awarded a contract with a standard price<BR>\n" );
document.write( "of at least RM250 million (before discounts).<BR>\n" );
document.write( "Company ABC offers a discount of RM10,000 per km of highway and RM60,000 per<BR>\n" );
document.write( "km of expressway. Meanwhile company RST offers a discount of RM20,000 per km<BR>\n" );
document.write( "of highway and RM50,000 per km of expressway\r<BR>\n" );
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document.write( "a. Construct the formulas for the problem above by using 𝑥 for highway and 𝑦 for<BR>\n" );
document.write( "expressway. Find the total discount to be named as D, received from both<BR>\n" );
document.write( "companies in Ringgit Malaysia.<BR>\n" );
document.write( "[15 marks]<BR>\n" );
document.write( "b. The corporate and strategic department targets to maximize the total discount,<BR>\n" );
document.write( "D. List out the SIX (6) constraints involved and evaluate the values of 𝑥 and 𝑦<BR>\n" );
document.write( "that will maximize D.<BR>\n" );
document.write( "[35 marks] </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850876 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=850876\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Absolutely, let's break down this problem into its components and solve it.\r<BR>\n" );
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document.write( "**a. Constructing the Formulas**\r<BR>\n" );
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document.write( "**Variables:**\r<BR>\n" );
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document.write( "* Let x_ABC = km of highway built by ABC<BR>\n" );
document.write( "* Let y_ABC = km of expressway built by ABC<BR>\n" );
document.write( "* Let x_RST = km of highway built by RST<BR>\n" );
document.write( "* Let y_RST = km of expressway built by RST\r<BR>\n" );
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document.write( "**Total Highway and Expressway Construction:**\r<BR>\n" );
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document.write( "* x_ABC + x_RST = 200 (Total highway)<BR>\n" );
document.write( "* y_ABC + y_RST = 100 (Total expressway)\r<BR>\n" );
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document.write( "**Cost (Standard Price):**\r<BR>\n" );
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document.write( "* Total cost = 1,000,000(x_ABC + x_RST) + 5,000,000(y_ABC + y_RST)<BR>\n" );
document.write( "* Total cost = 1,000,000(200) + 5,000,000(100) = 200,000,000 + 500,000,000 = RM 700,000,000\r<BR>\n" );
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document.write( "**Discount Formulas:**\r<BR>\n" );
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document.write( "* Discount from ABC = 10,000x_ABC + 60,000y_ABC<BR>\n" );
document.write( "* Discount from RST = 20,000x_RST + 50,000y_RST<BR>\n" );
document.write( "* Total Discount (D) = 10,000x_ABC + 60,000y_ABC + 20,000x_RST + 50,000y_RST\r<BR>\n" );
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document.write( "**Constraints:**\r<BR>\n" );
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document.write( "* ABC's Maximum Roadway: x_ABC + y_ABC ≤ 200<BR>\n" );
document.write( "* RST's Maximum Roadway: x_RST + y_RST ≤ 150<BR>\n" );
document.write( "* ABC's Minimum Contract Value: 1,000,000x_ABC + 5,000,000y_ABC ≥ 250,000,000 or x_ABC + 5y_ABC ≥ 250<BR>\n" );
document.write( "* RST's Minimum Contract Value: 1,000,000x_RST + 5,000,000y_RST ≥ 250,000,000 or x_RST + 5y_RST ≥ 250<BR>\n" );
document.write( "* x_ABC, y_ABC, x_RST, y_RST ≥ 0 (Non-negativity)\r<BR>\n" );
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document.write( "**b. Maximizing the Total Discount**\r<BR>\n" );
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document.write( "**Constraints:**\r<BR>\n" );
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document.write( "1.  x_ABC + x_RST = 200<BR>\n" );
document.write( "2.  y_ABC + y_RST = 100<BR>\n" );
document.write( "3.  x_ABC + y_ABC ≤ 200<BR>\n" );
document.write( "4.  x_RST + y_RST ≤ 150<BR>\n" );
document.write( "5.  x_ABC + 5y_ABC ≥ 250<BR>\n" );
document.write( "6.  x_RST + 5y_RST ≥ 250<BR>\n" );
document.write( "7. x_ABC, y_ABC, x_RST, y_RST >= 0\r<BR>\n" );
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document.write( "**Simplifying the Problem:**\r<BR>\n" );
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document.write( "* From (1), x_RST = 200 - x_ABC<BR>\n" );
document.write( "* From (2), y_RST = 100 - y_ABC\r<BR>\n" );
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document.write( "**Substituting into the Discount Formula:**\r<BR>\n" );
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document.write( "* D = 10,000x_ABC + 60,000y_ABC + 20,000(200 - x_ABC) + 50,000(100 - y_ABC)<BR>\n" );
document.write( "* D = 10,000x_ABC + 60,000y_ABC + 4,000,000 - 20,000x_ABC + 5,000,000 - 50,000y_ABC<BR>\n" );
document.write( "* D = -10,000x_ABC + 10,000y_ABC + 9,000,000\r<BR>\n" );
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document.write( "**Substituting into the Remaining Constraints:**\r<BR>\n" );
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document.write( "* (3) x_ABC + y_ABC ≤ 200<BR>\n" );
document.write( "* (4) (200 - x_ABC) + (100 - y_ABC) ≤ 150 => 300 - x_ABC - y_ABC ≤ 150 => x_ABC + y_ABC ≥ 150<BR>\n" );
document.write( "* (5) x_ABC + 5y_ABC ≥ 250<BR>\n" );
document.write( "* (6) (200 - x_ABC) + 5(100 - y_ABC) ≥ 250 => 200 - x_ABC + 500 - 5y_ABC ≥ 250 => x_ABC + 5y_ABC ≤ 450\r<BR>\n" );
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document.write( "**Revised Constraints:**\r<BR>\n" );
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document.write( "1.  x_ABC + y_ABC ≤ 200<BR>\n" );
document.write( "2.  x_ABC + y_ABC ≥ 150<BR>\n" );
document.write( "3.  x_ABC + 5y_ABC ≥ 250<BR>\n" );
document.write( "4.  x_ABC + 5y_ABC ≤ 450<BR>\n" );
document.write( "5. x_ABC, y_ABC >= 0\r<BR>\n" );
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document.write( "**Finding Corner Points:**\r<BR>\n" );
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document.write( "* x_ABC + y_ABC = 200<BR>\n" );
document.write( "* x_ABC + y_ABC = 150<BR>\n" );
document.write( "* x_ABC + 5y_ABC = 250<BR>\n" );
document.write( "* x_ABC + 5y_ABC = 450\r<BR>\n" );
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document.write( "**Solving the System of Equations:**\r<BR>\n" );
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document.write( "1.  **x_ABC + y_ABC = 200 and x_ABC + 5y_ABC = 450:**<BR>\n" );
document.write( "    * 4y_ABC = 250 => y_ABC = 62.5, x_ABC = 137.5<BR>\n" );
document.write( "2.  **x_ABC + y_ABC = 150 and x_ABC + 5y_ABC = 250:**<BR>\n" );
document.write( "    * 4y_ABC = 100 => y_ABC = 25, x_ABC = 125<BR>\n" );
document.write( "3.  **x_ABC + y_ABC = 200 and x_ABC + 5y_ABC = 250:**<BR>\n" );
document.write( "    * 4y_ABC = 50 => y_ABC = 12.5. x_ABC = 187.5<BR>\n" );
document.write( "4.  **x_ABC + y_ABC = 150 and x_ABC + 5y_ABC = 450:**<BR>\n" );
document.write( "    * 4y_ABC = 300 => y_ABC = 75. x_ABC = 75.\r<BR>\n" );
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document.write( "**Evaluating D:**\r<BR>\n" );
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document.write( "* D = -10,000x_ABC + 10,000y_ABC + 9,000,000\r<BR>\n" );
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document.write( "1.  (137.5, 62.5): D = -1,375,000 + 625,000 + 9,000,000 = 8,250,000<BR>\n" );
document.write( "2.  (125, 25): D = -1,250,000 + 250,000 + 9,000,000 = 8,000,000<BR>\n" );
document.write( "3. (187.5, 12.5): D=-1,875,000+125,000+9,000,000 = 7,250,000<BR>\n" );
document.write( "4. (75, 75): D = -750,000 + 750,000 + 9,000,000 = 9,000,000\r<BR>\n" );
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document.write( "**Maximum Discount:**\r<BR>\n" );
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document.write( "The maximum discount is RM 9,000,000 when x_ABC = 75 and y_ABC = 75.\r<BR>\n" );
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document.write( "* x_RST = 200 - 75 = 125<BR>\n" );
document.write( "* y_RST = 100 - 75 = 25\r<BR>\n" );
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document.write( "Therefore, ABC builds 75km of highway and 75km of expressway, and RST builds 125km of highway and 25km of expressway.<BR>\n" );
document.write( " </SPAN>\n" );
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