document.write( "Question 1131906: A bed manufacturer makes two types of bed: standard and luxury. The cost of manufacturing each type of bed is 2300$ for a standard model and 3700$ for a luxury model.
\n" ); document.write( "It costs 300$ to ship each standard model and 400$ to ship each luxury model. The maximum weekly costs are 851,000$ for manufacturing and 120,000$ for shipping. No more than 300 beds can be manufactured per week. How many bed of each type should be made to maximize profit if the profit is 6000$ on each standard and 8000$ on each luxury? \n" ); document.write( "

Algebra.Com's Answer #748649 by ikleyn(52781) $\"\"$ $\"About$

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document.write( "Let X = # of standard beds;\r\n" );
document.write( "    Y = # of luxury   beds.\r\n" );
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document.write( "The objective (profit) function is F(X,Y) = 6X + 8Y, in thousand dollars.\r\n" );
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document.write( "The constraints are\r\n" );
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document.write( "2.3X + 3.7Y <= 851      (1)    (maximum weekly cost, in thousand dollars)\r\n" );
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document.write( "0.3X + 0.4Y <= 120      (2)    (maximum weekly shipping, in thousand dollars)\r\n" );
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document.write( "X + Y <= 300            (3)\r\n" );
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document.write( "X >= 0,  Y >= 0.        (4)    (non-negativity)\r\n" );
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document.write( "The problem is to maximize the objective function under the given restrictions.\r\n" );
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document.write( "The feasible domain is shown in the Figure below.\r\n" );
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document.write( "It is a quadrilateral in QI under the red, green and blue lines - factually, under the red and blue lines.\r\n" );
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document.write( "Plot  2.3X + 3.7Y = 851 (red),  0.3X + 0.4Y = 120 (green)  and  X + Y <= 300 (blue)\r\n" );
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document.write( "The maximum (the solution to the problem) is achieved in one of the three corner points:\r\n" );
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document.write( "    P1 = (0,230)      (red line Y-intercept)\r\n" );
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document.write( "    P2 = (185,115)    (red line and blue line intersection point)\r\n" );
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document.write( "    P3 = (300,0)      (blue line X-intercept)\r\n" );
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document.write( "Now, calculate the value of the objective function at each of this three corner points\r\n" );
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document.write( "    at P1:  F(0,230)   = 6*0 + 8*230   = 1840 thousand dollars;\r\n" );
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document.write( "    at P2:  F(185,115) = 6*185 + 8*115 = 2030 thousand dollars;   and  \r\n" );
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document.write( "    at P3:  F(300,0)   = 6*300 + 8*0   = 1800 thousand dollars.\r\n" );
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document.write( "The maximum is achieved at P2, and this point gives the solution.\r\n" );
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document.write( "ANSWER.  The maximum profit is achieved when  185 standard beds and 115 luxury beds are produced per week.\r\n" );
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document.write( "         The maximum profit then is 2030 thousand dollars per week.\r\n" );
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\n" ); document.write( "\n" ); document.write( "To see other similar problems solved by the Linear Programming method, look into the lesson\r
\n" ); document.write( "\n" ); document.write( " - Solving minimax problems by the Linear Programming method \r
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\n" ); document.write( "\n" ); document.write( "Also, look into the solutions on other similar problems in the archive to this forum under the links\r
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\n" ); document.write( "\n" ); document.write( "https://www.algebra.com/algebra/homework/Graphs/Graphs.faq.question.1131906.html\r
\n" ); document.write( "\n" ); document.write( "https://www.algebra.com/algebra/homework/coordinate/word/Linear_Equations_And_Systems_Word_Problems.faq.question.1131043.html\r
\n" ); document.write( "\n" ); document.write( "https://www.algebra.com/algebra/homework/word/finance/Money_Word_Problems.faq.question.1129285.html\r
\n" ); document.write( "\n" ); document.write( "https://www.algebra.com/algebra/homework/Finance/Finance.faq.question.1128383.html\r
\n" ); document.write( "\n" ); document.write( "https://www.algebra.com/algebra/homework/Linear-equations/Linear-equations.faq.question.1123217.html\r
\n" ); document.write( "\n" ); document.write( "https://www.algebra.com/algebra/homework/Finance/Finance.faq.question.1102103.html\r
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