document.write( "Question 1122658: A small company builds two types of garden chairs.
\n" ); document.write( "Type A requires 2 hours of machine time and 5 hours of craftsman time.
\n" ); document.write( "Type B requires 3 hours of machine time and 5 hours of craftsman time.
\n" ); document.write( "Each day there are 30 hours of machine time available and 60 hours of craftsman time. The
\n" ); document.write( "profit on each type A chair is E60 and on each type B chair is E84. Formulate the appropriate
\n" ); document.write( "linear programming problem and solve it graphically to obtain the optimal solution that
\n" ); document.write( "maximizes profit
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Algebra.Com's Answer #738805 by ikleyn(52872) $\"\"$ $\"About$

You can put this solution on YOUR website!

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document.write( "Let X be the number of type A chairs;\r\n" );
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document.write( "let Y be the number of type B 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) = 60X + 84Y.\r\n" );
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document.write( "The restrictions are :\r\n" );
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document.write( "2X + 3Y <= 30       (1)     (restriction on the machine time)   and\r\n" );
document.write( "5X + 5Y <= 60       (2)     (restriction on the craftsman 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  2x + 3y = 30  and the green line  5X + 5Y = 60.\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) = (0,10)   (red line Y-intercept);\r\n" );
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document.write( "        (X2,Y2) = (6,6)   (intersection point of the straight lines Y =  and Y =  );\r\n" );
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document.write( "        (X3,Y3) = (12,0)   (green 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) = 60*0 + 84*10 =  840;\r\n" );
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document.write( "        R(X2,Y2) = 60*6 + 84*6  =  864;\r\n" );
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document.write( "        R(X3,Y3) = 60*12 + 84*0 =  720.\r\n" );
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document.write( "    3) Then select one of these point where the objective function is maximal - In our case this point is (X2,Y2) = (6,6).\r\n" );
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document.write( "    4) This point gives your optimal solution X = 6 chairs of type A and Y = 6 chairs of the type B.\r\n" );
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document.write( "If they follow this optimal solution, their weekly profit will be MAXIMAL, E864.\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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