Algebra.Com's Answer #755012 by ikleyn(52797)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "Let X be the number of tables, and\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) = 100X + 60Y. (1)\r\n" );
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document.write( "The restrictions are :\r\n" );
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document.write( "4X + 3Y <= 305 (2) (restriction on the assembly time) and\r\n" );
document.write( "2X + 1.5Y <= 355 (3) (restriction on the finishing time).\r\n" );
document.write( "X >= 0; Y >= 0. (4) (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 4X + 3Y = 305 and the green line 2X + 1.5Y = 355.\r\n" );
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document.write( "  \r\n" );
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document.write( " Plots 4X + 3Y = 305 (red) and 2X + 1.5Y = 355 (green)\r\n" );
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document.write( "Now I need to make couple of important notices.\r\n" );
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document.write( "1. The potential solutions are the points in this quadrilateral with integer coordinates X and Y.\r\n" );
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document.write( " Therefore, I show in the plot the grid of points with integer coordinates. But, due to technical restrictions, \r\n" );
document.write( " my grid is with the step 15 in both axes.\r\n" );
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document.write( " Actually, I'd should to show you the grid with the step 1 in both axes, but it would be not impossible to see on the screen.\r\n" );
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document.write( " Therefore my grid is with the step 15, and I ask you to stretch your imagination and to think that it is the grid with the step 1. \r\n" );
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document.write( "2. Next, from the plot you can see that working constrain is, actually, the green line.\r\n" );
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document.write( " The other, the red line, only constrains the set of possible solution, but does not work as real constrain in the search \r\n" );
document.write( " of the maximal profit.\r\n" );
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document.write( "3. The blue line in the plot is the line 100X + 60Y = const.\r\n" );
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document.write( " The left side represent the profit function, but it is not a real profit function: it is some its \"ghost\"/\"phantom\" - \r\n" );
document.write( " - its section by the plane P = const.\r\n" );
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document.write( " Actually, we should move this blue line PARALLEL TO ITSELF from the \"far outside\" area of quadrant QI closer \r\n" );
document.write( " and closer to the origin until this line touches for the first time a grid point in the feasible domain.\r\n" );
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document.write( " Then the value of the profit function at this point will be the maximal profit, and the point itself with its coordinates\r\n" );
document.write( " will give us the solution in terms of tables and chairs.\r\n" );
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document.write( "4. From the plot, it is clear that it will happen at high values of X close to x-intercept of the green line x=  = 76.25.\r\n" );
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document.write( " The simplest way to find this value of X is to express Y =  from (2) and then find its \r\n" );
document.write( " integer solution in X and Y in that domain.\r\n" );
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document.write( " It can be done MOMENTARILY (I used Excel, and it gave me integer solution (X,Y) = (74,3) ).\r\n" );
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document.write( "Thus the optimal solution is the point X = 74 tables and Y = 3 chairs.\r\n" );
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document.write( "If they follow this optimal solution, their weekly profit will be MAXIMAL, P(X,Y) = 100*X + 60*Y = 100*74 + 60*3 = 7580.\r\n" );
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document.write( "Solved.\r
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document.write( "For similar solved problem see the lesson\r
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document.write( " - Solving minimax problems by the Linear Programming method, Problem 6\r
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document.write( "in this site.\r
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document.write( "From this lesson, learn the technique and the methodology of solving minimax problems using the Linear Programming method.\r
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document.write( "Another way to solve this problem is to find (free of charge, preferably) Internet site/solver for solving
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document.write( "Linear Programming problems.\r
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document.write( "Then all you need is to input the setup equations and inequalities into the solver and press the \"Solve\" button.\r
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document.write( "If you do it, do not forget to inform the solver that you are looking for integer solutions.\r
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document.write( "But in any case, you have the setup from me together with detailed explanation on how the method works.\r
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