Algebra.Com's Answer #825163 by ikleyn(52788)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "A candy manufacturer makes two types of special candy, say A and B.
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document.write( "Candy A consists of equal part dark chocolate and caramel and
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document.write( "Candy B consists of two parts of dark chocolate and one part of walnut.
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document.write( "The company has in stock 430 kilograms of caramel, 360 kilograms of dark chocolate and 210 kilograms of walnuts.
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document.write( "The company sells Candy A for P285 and Candy B for P260 per kilogram.
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document.write( "How much of each candy should the manufacturer produce to maximize  revenue ?
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document.write( " First, the question is posed incorrectly.\r
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document.write( " The correct question should ask about maximizing the revenue (not the profit).\r
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document.write( " So, I will solve the problem in this edited/modified formulation.\r
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document.write( "Let x be the amount of candy A, in kilograms;\r\n" );
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document.write( "let y be the amount of candy B.\r\n" );
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document.write( "The revenue function (the subject to maximize) is R(x,y) = 285x + 260y. (1)\r\n" );
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document.write( "The constraint inequalities are\r\n" );
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document.write( "  +  <= 360 (2) (dark chocolate)\r\n" );
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document.write( " <= 430 (3) (caramel)\r\n" );
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document.write( "  <= 210 (4) (walnuts)\r\n" );
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document.write( " x >= 0, y >= 0.\r\n" );
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document.write( "Constraints can be rewritten equivalently\r\n" );
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document.write( " 3x + 4y <= 2160 (2')\r\n" );
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document.write( " x <= 860 (3')\r\n" );
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document.write( " y <= 630 (4')\r\n" );
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document.write( "The feasibility domain is shown in the plot below.\r\n" );
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document.write( "  \r\n" );
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document.write( " Figure. Plots 3x + 4y = 2160 (red); x = 860 (blue); y = 630 (green)\r\n" );
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document.write( "The feasibility domain is the triangle in QI, adjacent to x- and y-axis below the red line.\r\n" );
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document.write( "It is clear from the plot, that the real working constraint is (2').\r\n" );
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document.write( "The constraints (3') and (4') do not work.\r\n" );
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document.write( "In other words, the only working constraint is (2) for dark chocolate.\r\n" );
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document.write( "The other constraints (3) and (4) do not work.\r\n" );
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document.write( " | It means, that the solution and the strategy in this problem is to use |\r\n" );
document.write( " | as much dark chocolate as possible (as constraint (2') or (2) allow it). |\r\n" );
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document.write( "Now, to get the solution in x and y variables, you have a linear function of the revenue (1)\r\n" );
document.write( "over the triangle, which is the feasibility domain.\r\n" );
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document.write( "As the Linear Programming method says (and it is obvious from the common sense of view),\r\n" );
document.write( "the solution is one of the two vertices of the triangle, \r\n" );
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document.write( " (x,y) = ( , ) = (0,540) or (x,y) = ( ,) = (720,0).\r\n" );
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document.write( "So, you calculate the revenue at these two points\r\n" );
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document.write( " R(0,540) = 285*0 + 260*540 = 140400\r\n" );
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document.write( " R(720,0) = 285*720 + 260*0 = 205200.\r\n" );
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document.write( "Comparing it, you see that the optimum solution is to produce and to sell 720 kilograms of candy A and 0 kg of candy B,\r\n" );
document.write( "which gives the maximum revenue of P205200. ANSWER\r\n" );
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document.write( "Solved.\r
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document.write( "To see many other similar and different mini-max problems solved by the Linear Programming method, look into the lesson\r
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document.write( " - Solving minimax problems by the Linear Programming method \r
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document.write( "in this site.\r
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