document.write( "Question 1123217: The Alpha Furniture Company manufactures dining room tables and chairs. Each table requires 50 board feet and 3 hours of labor. Each chair takes 20 board feet and 4 hours of labor. The manufacturer has 3,300 board feet of lumber available and a staff able to provide 380 hours of labor. Finally, the manufacturer has determined that there is a net profit of P100 for each chair sold and P200 for every table sold. For simplicity, we assume that needed materials (such as nails and varnish) are available in sufficient quantities. How many tables and chairs should the company manufacture in order to maximize its profits, assuming that each item manufactured is sold? \n" ); document.write( "

Algebra.Com's Answer #739573 by ikleyn(52787) $\"\"$ $\"About$

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document.write( "Let X be the number of tables;\r\n" );
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document.write( "let Y be the number of chairs.\r\n" );
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document.write( "The objective function (profit)  is\r\n" );
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document.write( "R(X,Y) = 200X + 100Y.\r\n" );
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document.write( "The restrictions are :\r\n" );
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document.write( "50X + 20Y <= 3300       (1)     (restriction on board feet of lumber)   and\r\n" );
document.write( " 3X +  4Y <=  380       (2)     (restriction on labor time).\r\n" );
document.write( "X >= 0;  Y >= 0.        (3)     (non-negativity).\r\n" );
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document.write( "You need to maximize the objective function (profit) under given restrictions.\r\n" );
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document.write( "The feasible domain is shown below.\r\n" );
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document.write( "It is  a quadrilateral in the first quadrant  (X >= 0,  Y >= 0)  restricted \r\n" );
document.write( "by the red line  50X + 20Y = 3300  and the green line  3X + 4Y = 380.\r\n" );
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document.write( "Plots Y =   (red) and y =  (green)\r\n" );
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document.write( "The method of linear programming says:\r\n" );
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document.write( "    1) Take the vertices of this quadrilateral\r\n" );
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document.write( "        (X1,Y1) = (0,95)    (green line Y-intercept);\r\n" );
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document.write( "        (X2,Y2) = (40,65)   (intersection point of the straight lines Y =  and Y =  );\r\n" );
document.write( "        (X3,Y3) = (66,0)    (red line X-intercept)\r\n" );
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document.write( "    2) Calculate the objective function at these points\r\n" );
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document.write( "        R(X1,Y1) = 200*0 + 100*95   =  9500;\r\n" );
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document.write( "        R(X2,Y2) = 200*40 + 100*65  =  14500;\r\n" );
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document.write( "        R(X3,Y3) = 200*66 + 100*0   =  13200.\r\n" );
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document.write( "    3) Then select one of these points where the objective function is maximal - In our case this point is (X2,Y2) = (40,65).\r\n" );
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document.write( "    4) This point gives your optimal solution X = 40 tables  and Y = 65 chairs.\r\n" );
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document.write( "If they follow this optimal solution, their profit will be MAXIMAL, $14500.\r\n" );
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\n" ); document.write( "\n" ); document.write( "For many other similar solved problems see 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( "Learn from there the technique and the methodology on how to solve minimax problems using the Linear Programming method - once and for all.\r
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